Rodero Talk4

8/10/2019 Rodero Talk4

1/44

Superconduc t ing transpo r t

Superconducting model Hamiltonians:

Nambu formalismCurrent through a N/S junction

Supercurrent in an atomic contact

Finite bias current and shot noise:The MAR mechanism


2/44

Superconduct ing model Ham i l tonians

Ass ume an electron ic sy stem w ith Ham i l ton ian

(in a s ite representation ):

)(t iii i

iii

ccccnH

110

If due to some attract ive interact ion non includ ed in H, the sys tem

becomes superconduct ing :

i

iiiiii

i i

iiiS )()(t ccccccccnH

110

t0 0 0 0 0

t t t

= local pairing potential = gap parameter (homogeneous system)

ii

ii

cc

cc 0


3/44

t0 0 0 0 0t t t

i

iiiiii

i i

iiiS )()(t ccccccccnH

110

Diagonal izat ion of HS: Bogol iubo v transform at ion:

iiiii

iiiii

vu

vu

cc

cc

A quasi-part ic le is a l inear combinat ion o f electron and hole

2x2 space (Nambu space)


4/44

Matr ix notat ion: sp inor op erator for a quasi part ic le of sp in

i

i

ic

c iii cc

The usual causal propagator in this 2X2 space wil l be

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

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

jiji

jiji

ijccTccT

ccTccTG

Whic h in an exp l ic i t 2x2 representat ion has the form

)'t()t(i)'t,t( iiij TG


5/44

From a pract ical po int of view of the quantum mechanical calculat ion:

Doub l ing up of the Hi lbert space:

t0 0 0 0 0

t t t

0

0

0

h

tt0

0t

0

0

0

0

t

t

0

0

Formal ly l ike a normal sys tem with tw o o rb i tals per si te


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Problem:su r face Green fun ct ions in the superconduct ing state

th0 h0 h0 h0 h0t t t

Simple model: sem i-inf in i te t ight-bindin g chain

t0 0 0 0

t t

1234

su rface site

0

0

0

0

0

h

t

t

0

0t

e-h s ymmetry

00


7/44

Adding an extra ident ical si te, , and solv ing the Dyson equation0

010002

00

2 )(g)()(gt Normal case

00002

00 IghItg )()()( Supercond uct ing c ase

In a superconducto r the energies of in terest are

Wide band approximat ion

W

i)(i)(g 00 Normal state

2200

1)(i)(g Superconduct ing state

BCS density of states


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A word on notation: Nambu space+ Keldish space

Superconduct iv i ty Non-equi l ibr ium

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

21,j,i

Keld ish

Nambu


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N/S superconduc t ing con tact

Single-channel model

)(t LRRLRL

ccccHHH per turbat ion

L R

tLeft lead Righ t lead

eVRL Superconductor


10/44

Supercond uct in g r ight lead (un cou pled):

R

22

1)(i)( R

a

RRg

)(f)()()( Rr

RR

a

RR

,

RR ggg

0R

Nambu s pace


11/44

Norm al metal left lead

1001)(i)( L

a

LL g

L

)(f)()()( Lr

LL

a

LL

,

LL ggg )eV(f)(fL

)eV(f

)eV(f)(i)( L

,

LL

0

02g

hole dist r ibut ion

Impor tant point


12/44

I

V

12

eV

0T

0T

N/S quasi-particle tunnel: tunnel l imi t

Differential conductance

standard BCS picture

)(

)eV(

G

)V(G

N

S

N

S

eV,)eV(

eV

22

eV,0


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-3 -2 -1 0 1 2 3

0

1

2

G(V)/G

0

eV/

= 1= 0.9= 0.5

)exp( dt

dTunnel r egime

Contact regime

0

1

h

eGG

2

0

42

eV

onductance saturation

1


14/44

Normal metal Superconductor

And reev Reflect ion

Probability2Transmitted charge e2


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)(t LRRLRL

ccccHHH

per turbat ion

)(G)(Gdth

e ,,LR

,

,RL 1111

2I

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

ccccI

)t()t()t()t(tieLRRL

ccccI

2

L R


eVRL

SuperconductorNormal metal


16/44

Current due to Andreev ref lect ions (eV

][)(

8 2

122211

42

)eV(f)eV(fG)eV()eV(dth

e

)V(I ,S,M,MA

)eV(,M 222

12 )(,SG)eV(,M 11


17/44

h

eG

2

0

2

-3 -2 -1 0 1 2 3

0

1

2

G(V)/G

0

eV/

= 1= 0.9= 0.5

Differential conductance

)/eV)(()(h

e)V(G

142

42

22

eV

h

e)V(G

24 1saturat ion value


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Josephson current in a S/S contact

Zero bias case

L R


0 RL

SuperconductorSuperconductor

Superconduct ing phase dif ference

RLLi

L e RiR e

)(t LRRLRL

ccccHHH

BCS sup ercondu ctors


19/44

12

SQUID configuration

transmission

L

Li

L e

L

L

i

i

L

a

LLe

e)(i)(

22

1g

Nambu space

Uncoup led superconductors


20/44

)(t LRRLRL

ccccHHH

per turbat ion

)(G)(Gdth

e ,,LR

,

,RL 1111

2I

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

ccccI

)t()t()t()t(tieLRRL

ccccI

2

L R

t

Left lead Righ t lead

0 RL



21/44

)(G)(Gdth

e)(I , ,LR

,

,RL 1111

2

The zero b ias case, V=0, is special ly s imple, becaus e the sys tem

is in equi l ibr ium

Even in the per turbed s ystem:

)(f)()()( ra,

GGG

)(f)(G)(G)(G r ,RLa

,RL

,

,RL 111111

) ) )(fGGGGdth

e)(I r ,LR

r

,RL

a

,LR

a

,RL 111111112


22/44

) ) )(fGGGGdth

e)(I r ,LR

r

,RL

a

,LR

a

,RL 111111112

)(f)(D

)(g)(gImdsint

h

e)(I

r

,R

r

,L

21122

2

1)(D Tunnel l imit

Tktanhsin

eR)(I

BN 22

Ambegaokar-Baratof f

][ )(gt)(tgdet)(D rRrL I

222112

)i()(g)(g rr

Nambu sp ace

)()(


23/44

-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

21122

2

0)(DAndreev states

2

1 2 sin)(


24/44

)(f)(D

)(g)(gImdsint

h

e)(I

21122

2

0)(D

2

1 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

he)(I

B

s2

2

Supercurrent

d

)(de)(IS

Two level sys tem


25/44

Josephson supercurrent

21

2 2

sen

sene)(Is

0

senh

eIs

)(

1 22)( sen

h

eIs

Jo seph son (1962)

Ku lik-Omelyanch uk (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


26/44

S/S atomic contact with f ini te bias

Mul tiple Andreev refl ections (MAR)

Sub-gap structure:quali tat ive explanation

e

a) 1 quasi-particleeV>

1

p

e

h

b) eV>

2

2 p

e

e

h

c) 3 quasi-particleseV>23

3

3 p

2 quasi-particles

I

V

a

b

3

c

nquasi-particleseV>2n


27/44

Conduct ion in a supercondu ct ing junct ion

2 2

I

eV2

EF,L

EF,L - EF,R = eV > 2

2

EF,R

I


28/44

Experimental IVcurves in superconduc t ing contacts

0 100 200 300 400 5000

10

20

30

40

50

T = 17 mK

V [ V ]

I[nA

]

3Al 1 atom

contact


29/44

Superconductor

Superconductor

Andreev ref lect ion in a superconduct ing jun ct ion

eV>

I

eV2

Probability

2

Transmitted charge e2


30/44

Th ti l d l


31/44

Theoretical model

eVRL

eV

dt

d 2

t

eV

t

2

)( 0

2/)(

2/)(

ti

R

ti

L

e

e

2/)t(itet Gauge cho ice

V

n

tin

n eVItVI

)()(),(

][

LR

)t(i

RL

)t(i

RL tete ccccHHH

t ime d ependent per turbat ion

L R


eVRL



32/44

dc com ponent of the current I0(V)

Calcu lat ion of th e 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 1121122

n

)t(in

n e)V(I)t,V(I

Non- l inear and non-stat ion ary current

Experiments

][

LR

)t(i

RL

)t(i

RL tete ccccHHH


33/44

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5 TRANSMISSION1.0

0.99

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

eV/

eI/G

Theoretical IVcurves

3


34/44

0 100 200 300 400 5000

10

20

30

40

50

T = 17 mK

V [ V ]

I[nA

]

3

Alone-atom contact

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

dc current

TRANSMISSION

1.00.99

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

eV/

eI/G

Sub-gap str uc ture (SGS) in:n

Ve 2


35/44

0 1 2 3 4 5 6

0

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


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


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


eI/G

eV/

Fit t ing of the cu rves I0(V)


36/44

I0(V) character istics

0 1 2 30

1

2

3

4T

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

0.875

0.817

0.807

eV/

eI/G

Atomic Al contacts

0 1 2 3 4 50

2

4

edc

ba

eI/G

eV/

Atom ic Pb contacts


37/44

Mechanical break jun ct ion

Superconduct ingIV in last con tact before breaking

Theoret ical cu rves

Determination of conduction channels of an atomic contact

Scheer et al, PRL 78, 3535 (97)

(Saclay)

n


38/44

The PIN code of an atom ic contact

n

nh

eG

22

PIN coden


39/44

Correlat ion between num ber of channels and num ber of valence atomic o rbi ta ls

3s

3pAl

eV7~

A l 3Pb 3Nb 5Au 1

(Saclay)

(Saclay)

(Leiden)

(Madrid)

MCBJ

MCBJ

MCBJ

STM

Proximity effect

Determination of conduction channels of an atomic contact


40/44

Shot noise in superconducting atomic contacts

TkeV B

eIS 2)0( Poisson ian l imi t

*2/)0( qIS Charge of the carriers

)t()()()t(dt)(S IIII 000 0

What is the transm it ted charge in a Andreev ref lect ion?

e

eV>

e

h

eV>

e

e

h

eV>23

eQ * eQ 2* eQ 3* ? ?


41/44

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

7

8 0.95

Shot Noise

0.9

0.8 0.7

0.60.5

0.40.3

0.2

0.1

1.0

S/(4e

2

/h)

eV/

Huge increase of S/2eI for V 0

Theoretical curves


42/44

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/2eI

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/

Effect ive ch arge carr ied b y a m ult ip le Andreev ref lect ion:

eVQ

2Integer1*


43/44

Shot noise measurements in atomic contacts

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

superconduct ing Al con tac t

effective charge


44/44

SC SC

FS S

Supercond uct ing transpor t throug h a magnet ic region

Superconduct ing transp or t through a correlated quantum dot

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