Superconducting transport

Superconducting model Hamiltonians: Nambu formalism Current through a N/S junction Supercurrent in an atomic contact Finite bias current and shot noise: The MAR mechanism

Superconducting model Hamiltonians

Assume an electronic system with Hamiltonian

(in a site representation):

)(t iii i

iii

ccccnH

110

If due to some attractive interaction non included in H, the system

becomes superconducting:

i

iiiiiii i

iiiS )()(t ccccccccnH

110

t0 0 0 0 0t t t

= local pairing potential = gap parameter (homogeneous system)

ii

ii

cc

cc 0

t0 0 0 0 0t t t

i

iiiiiii i

iiiS )()(t ccccccccnH

110

Diagonalization of HS: Bogoliubov transformation:

iiiii

iiiii

vu

vu

ccγ

ccγ

A quasi-particle is a linear combination of electron and hole

2x2 space (Nambu space)

Matrix notation: spinor operator for a quasi particle of spin

i

ii c

cψ

iii ccψ

The usual causal propagator in this 2X2 space will be

)'t()t()'t()t(

)'t()t()'t()t(i)'t,t(

jiji

jijiij

ccTccT

ccTccTG

Which in an explicit 2x2 representation has the form

)'t()t(i)'t,t( iiij ψψTG

From a practical point of view of the quantum mechanical calculation:

Doubling up of the Hilbert space:

t0 0 0 0 0t t t

0

00

h

t

t

0

0t

0

0

0

0

t

t

0

0

Formally like a normal system with two orbitals per site

Problem: surface Green functions in the superconducting state

th0 h0 h0 h0 h0t t t

Simple model: semi-infinite tight-binding chain

t0 0 0 0

t t

1234

surface site

0

0

0

00

h

t

t

0

0t

e-h symmetry

00

Adding an extra identical site, , and solving the Dyson equation0

01000200

2 )(g)()(gt Normal case

00002

00 IghItg )()()( Superconducting case

In a superconductor the energies of interest are

Wide band approximation

W

i)(i)(g 00 Normal state

2200

1)(i)(g Superconducting state

BCS density of states

A word on notation: Nambu space + Keldish space

Superconductivity Non-equilibrium

)'t,t(G ,j,i ,,

21,j,i

Keldish

Nambu

N/S superconducting contact

Single-channel model

)(t LRRLRL

ccccHHH perturbation

L R

tLeft lead Right lead

eVRL Superconductor

Superconducting right lead (uncoupled):

R

22

1)(i)( R

aRRg

)(f)()()( RrRR

aRR

,RR ggg

0R

Nambu space

Normal metal left lead

10

01)(i)( L

aLL g

L

)(f)()()( LrLL

aLL

,LL ggg )eV(f)(fL

)eV(f

)eV(f)(i)( L

,LL

0

02g

hole distribution

Important point

I

V

12

eV0T

0T

N/S quasi-particle tunnel: tunnel limit

Differential conductance

standard BCS picture

)(

)eV(

G

)V(G

N

S

N

S

eV,)eV(

eV22

eV,0

-3 -2 -1 0 1 2 30

1

2

G(V

)/G

0

eV/

= 1 = 0.9 = 0.5

)exp( dt

dTunnel regime

Contact regime

0

1

h

eGG

2

0

42

eV

Conductance saturation

1

Normal metal Superconductor

Andreev Reflection

Probability 2Transmitted charge e2

)(t LRRLRL

ccccHHH perturbation

)(G)(Gdth

e ,,LR

,,RL 1111

2I

)t()t()t()t(tie

LRRL

ccccI

)t()t()t()t(tie

LRRL

ccccI

2

L R

tLeft lead Right lead

eVRL

SuperconductorNormal metal

Current due to Andreev reflections (eV

][)(8 2

12221142 )eV(f)eV(fG)eV()eV(dt

h

e)V(I ,S,M,MA

)eV(,M 22

2

12 )(,SG)eV(,M 11

h

eG

2

0

2

-3 -2 -1 0 1 2 30

1

2

G(V

)/G

0

eV/

= 1 = 0.9 = 0.5

Differential conductance

)/eV)(()(h

e)V(G

142

42

22eV

h

e)V(G

24 1saturation value

Josephson current in a S/S contact

Zero bias case

L R

tLeft lead Right lead

0 RL SuperconductorSuperconductor

Superconducting phase difference

RLLi

L e RiR e

)(t LRRLRL

ccccHHH

BCS superconductors

12

SQUID configuration

transmission

L

LiL e

L

L

i

i

LaLL

e

e)(i)(

22

1g

Nambu space

Uncoupled superconductors

)(t LRRLRL

ccccHHH perturbation

)(G)(Gdth

e ,,LR

,,RL 1111

2I

)t()t()t()t(tie

LRRL

ccccI

)t()t()t()t(tie

LRRL

ccccI

2

L R

tLeft lead Right lead

0 RL

SuperconductorSuperconductor

)(G)(Gdth

e)(I ,

,LR,,RL 1111

2

The zero bias case, V=0, is specially simple, because the system is in equilibrium

Even in the perturbed system:

)(f)()()( ra, GGG

)(f)(G)(G)(G r,RL

a,RL

,,RL 111111

)(fGGGGdth

e)(I r

,LRr

,RLa

,LRa

,RL 11111111

2

)(fGGGGdth

e)(I r

,LRr

,RLa

,LRa

,RL 11111111

2

)(f)(D

)(g)(gImdsint

h

e)(I

r,R

r,L

211222

1)(D Tunnel limit

Tktanhsin

eR)(I

BN 22 Ambegaokar-Baratoff

][ )(gt)(tgdet)(D rR

rL I

222112)i(

)(g)(g rr

Nambu space

-3 -2 -1 0 1 2 30

1

2

= 0.1

-3 -2 -1 0 1 2 30

1

2

3

4

5

= 0.95

-3 -2 -1 0 1 2 3-30

-20

-10

0

10

20

30j()

= 0.95 = 2.5

)(f)(D

)(g)(gImdsint

h

e)(I

211222

0)(D Andreev states

21 2 sin)(

)(f)(D

)(g)(gImdsint

h

e)(I

211222

0)(D

21 2 sin)(Andreev states

-2 -1 0 1 2-1.0

-0.5

0.0

0.5

1.0

=0.9

E/

/

Tk

)(tanh)(

sen

h

e)(I

Bs 2

2

Supercurrent

d

)(de)(IS Two level system

Josephson supercurrent

21

2 2

sen

sene)(I s

0 senh

eI s

)(

1 2

2)(

senh

eI s

Josephson (1962)

Kulik-Omelyanchuk (1977)

0,0 0,5 1,0 1,5 2,0

-0,10

-0,05

0,00

0,05

0,10

I()/Ic

= 0.1

0,0 0,5 1,0 1,5 2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5I()/Ic

=0.9

0,0 0,5 1,0 1,5 2,0

-2

-1

0

1

2I()/I

c

=1

S/S atomic contact with finite bias

Multiple Andreev reflections (MAR)

Sub-gap structure: qualitative explanation

e

a) 1 quasi-particleeV>

1p

e

h

b) eV>

2

2 p

e

eh

c) 3 quasi-particleseV>2

3

3 p

2 quasi-particles

I

V

a

b

c

n quasi-particleseV>2n

Conduction in a superconducting junction

2 2

I

eV2

EF,L

EF,L - EF,R = eV > 2

2EF,R

I

Experimental IV curves in superconducting contacts

0 100 200 300 400 5000

10

20

30

40

50

T = 17 mK

V [ µV ]

I [ n

A ]

Al 1 atomcontact

Superconductor

Superconductor

Andreev reflection in a superconducting junction

eV>

I

eV2

Probability 2Transmitted charge e2

Superconductor

Superconductor

Multiple Andreev reflection

eV > 2/3

I

eV22 /3

Probability 3Transmitted charge e3

Theoretical model

eVRL eV

dt

d 2

teV

t

2)( 0

2/)(

2/)(

ti

R

ti

L

e

e

2/)t(itet Gauge choice

V

n

tin

n eVItVI )()(),(

][

LR)t(i

RL)t(i

RL tete ccccHHH time dependent perturbation

L R

tLeft lead Right lead

eVRL

SuperconductorSuperconductor

dc component of the current I0(V)

Calculation of the current

][ 22 )t(c)t(cte)t(c)t(cteie

)t(I LR/)t(i

RL/)t(i

)t,t(Gte)t,t(Gtee

)t(I ,LR/)t(i

,RL/)t(i 11

211

22

n

)t(inn e)V(I)t,V(I

Non-linear and non-stationary current

Experiments

][

LR)t(i

RL)t(i

RL tete ccccHHH

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5 TRANSMISSION 1.0 0.99 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

eV/

eI/G

Theoretical IV curves

0 100 200 300 400 5000

10

20

30

40

50

T = 17 mK

V [ µV ]

I [ n

A ]

Al “one-atom” contact

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

dc current

TRANSMISSION 1.0 0.99 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

eV/eI

/G

• Sub-gap structure (SGS) in:n

Ve

2

0 1 2 3 4 5 60

1

2

3

4

5

experimental data(Total transmission = 0.807)

eI/G

eV/0 1 2 3 4 5 6

0

1

2

3

4

5

n = 0.652

experimental data(Total transmission = 0.807)

eI/G

eV/0 1 2 3 4 5 6

0

1

2

3

4

5

n = 0.652

n = (0.390,0.388)

experimental data(Total transmission = 0.807)

eI/G

eV/0 1 2 3 4 5 6

0

1

2

3

4

5

n = 0.652

n = (0.390,388)

n = (0.405,0.202,0.202)

experimental data(Total transmission = 0.807)

eI/G

eV/

Fitting of the curves I0(V)

I0(V) characteristics

0 1 2 30

1

2

3

4 T

1=0.800, T

2=0.075

T1=0.682, T

2=0.120, T

3=0.015

T1=0.399, T

2=0.254, T

3=0.154

eV/

eI/G

Atomic Al contacts

0 1 2 3 4 50

2

4

edc

ba

eI/G

eV/

Atomic Pb contacts

Mechanical break junction

Superconducting IV in last contact before breaking

Theoretical curves

Determination of conduction channels of an atomic contact

Scheer et al, PRL 78, 3535 (97)(Saclay)

n

The PIN code of an atomic contact

n

nh

eG

22PIN code n

Correlation between number of channels and number of valence atomic orbitals

3s

3pAl

eV7~

• Al 3• Pb 3• Nb 5• Au 1

(Saclay)

(Saclay)

(Leiden)(Madrid)

MCBJ

MCBJ

MCBJSTM

Proximity effect

Determination of conduction channels of an atomic contact

Shot noise in superconducting atomic contacts

TkeV B

eIS 2)0( Poissonian limit

*2/)0( qIS Charge of the carriers

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

0

What is the transmitted charge in a Andreev reflection?

e

eV>

e

h

eV>

e

eh

eV>2

eQ * eQ 2* eQ 3* ? ?

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

7

80.95

Shot Noise

0.9

0.80.7

0.6 0.5

0.40.3

0.2

0.11.0

S/(4

e2

/h)

eV/• Huge increase of S/2eI for V 0

Theoretical curves

0,5 1,0 1,5 2,0 2,5 3,00

1

2

3

4

5

Charge in the tunnel limit

= 0.01

= 0.1

S/2e

I

eV/0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

0

2

4

6

8

10

Effective charge

Transmission 0.2 0.4 0.6 0.8 0.95q

= S/

2eI

eV/

Effective charge carried by a multiple Andreev reflection:

eV

Q2

Integer1*

Shot noise measurements in atomic contacts

• Cron, Goffman, Esteve and Urbina, Phys.Rev.Lett. 86, 4104, (2001).

superconducting Al contact

effective charge

SC SC

FS S

Superconducting transport through a magnetic region

Superconducting transport through a correlated quantum dot