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15 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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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
i
i
i
i
i
i
i








c
c
c
c
n
H









1
1
0


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


becomes superconducting:





















i
i
i
i
i
i
i
i
i
i
i
i
S
)
(
)
(
t
c
c
c
c
c
c
c
c
n
H








1
1
0
t


0


0


0


0


0

t

t

t













=
local pairing potential = gap parameter (homogeneous system)








i
i
i
i
c
c
c
c
0
t


0


0


0


0


0

t

t

t






























i
i
i
i
i
i
i
i
i
i
i
i
S
)
(
)
(
t
c
c
c
c
c
c
c
c
n
H








1
1
0


Diagonalization of
H
S
: Bogoliubov transformation:















i
i
i
i
i
i
i
i
i
i
v
u
v
u
c
c
γ
c
c
γ


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
i
i
c
c
ψ







i
i
i
c
c
ψ


The usual causal propagator in this 2X2 space will be
































)
'
t
(
)
t
(
)
'
t
(
)
t
(
)
'
t
(
)
t
(
)
'
t
(
)
t
(
i
)
'
t
,
t
(
j
i
j
i
j
i
j
i
ij
c
c
T
c
c
T
c
c
T
c
c
T
G


Which in an explicit 2x2 representation has the form




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



ψ
ψ
T
G


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




Doubling up of the Hilbert space:

t


0


0


0


0


0

t

t

t













0
0
0


h










t
t
0
0
t











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

t

h
0

h
0

h
0

h
0

h
0

t

t

t

Simple model: semi
-
infinite tight
-
binding chain

t


0


0


0


0

t

t

1
2
3
4
surface site
























0
0
0
0
0


h










t
t
0
0
t
e
-
h symmetry

0
0






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

0
0
1
00
0
2
00
2




)
(
g
)
(
)
(
g
t




Normal case



0
00
0
2
00




I
g
h
I
tg
)
(
)
(
)
(



Superconducting case



In a superconductor the energies of interest are




Wide band approximation

W
i
)
(
i
)
(
g





00
Normal state






















2
2
00
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





,
,


2
1
,
j
,
i

Keldish

Nambu

N/S superconducting contact

Single
-
channel model

)
(
t
L
R
R
L
R
L





c
c
c
c
H
H
H







perturbation

L

R

t

Left lead

Right lead

eV
R
L




Superconductor



Superconducting right lead (uncoupled):

R






















2
2
1
)
(
i
)
(
R
a
RR
g


)
(
f
)
(
)
(
)
(
R
r
RR
a
RR
,
RR




g
g
g




0

R

Nambu space



Normal metal left lead










1
0
0
1
)
(
i
)
(
L
a
LL



g
L



)
(
f
)
(
)
(
)
(
L
r
LL
a
LL
,
LL




g
g
g




)
eV
(
f
)
(
f
L

















)
eV
(
f
)
eV
(
f
)
(
i
)
(
L
,
LL






0
0
2
g
hole distribution



Important point

I

V



1

2

eV
0
T

0
T


N/S quasi
-
particle tunnel: tunnel limit

Differential conductance

standard BCS picture

)
(
)
eV
(
G
)
V
(
G
N
S
N
S









eV
,
)
eV
(
eV
2
2



eV
,
0
-3
-2
-1
0
1
2
3
0
1
2
G(V)/G
0
eV/



= 1


= 0.9


= 0.5
)
exp(
d
t



d

Tunnel regime

Contact regime

0


1


h
e
G
G
2
0
4
2




eV
Conductance saturation

1


Normal metal

Superconductor

Andreev Reflection

Probability

2


Transmitted charge

e
2
)
(
t
L
R
R
L
R
L





c
c
c
c
H
H
H







perturbation



)
(
G
)
(
G
d
t
h
e
,
,
LR
,
,
RL










11
11
2
I


)
t
(
)
t
(
)
t
(
)
t
(
t
ie
L
R
R
L





c
c
c
c
I








)
t
(
)
t
(
)
t
(
)
t
(
t
ie
L
R
R
L








c
c
c
c
I

2
L

R

t

Left lead

Right lead

eV
R
L




Superconductor

Normal metal

Current due to Andreev reflections (eV

)

]
[
)
(
8
2
12
22
11
4
2
)
eV
(
f
)
eV
(
f
G
)
eV
(
)
eV
(
d
t
h
e
)
V
(
I
,
S
,
M
,
M
A















)
eV
(
,
M



22
2
12
)
(

,
S
G
)
eV
(
,
M



11
h
e
G
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








1
4
2
4
2
2
2


eV
h
e
)
V
(
G
2
4

1


saturation value

Josephson current in a S/S contact

Zero bias case

L

R

t

Left lead

Right lead

0


R
L


Superconductor

Superconductor

Superconducting phase difference






R
L
L
i
L
e




R
i
R
e




)
(
t
L
R
R
L
R
L





c
c
c
c
H
H
H







BCS superconductors



1
2






SQUID configuration

t
ransmission

L

L
i
L
e




























L
L
i
i
L
a
LL
e
e
)
(
i
)
(
2
2
1
g
Nambu space

Uncoupled superconductors

)
(
t
L
R
R
L
R
L





c
c
c
c
H
H
H







perturbation



)
(
G
)
(
G
d
t
h
e
,
,
LR
,
,
RL










11
11
2
I


)
t
(
)
t
(
)
t
(
)
t
(
t
ie
L
R
R
L





c
c
c
c
I








)
t
(
)
t
(
)
t
(
)
t
(
t
ie
L
R
R
L








c
c
c
c
I

2
L

R

t

Left lead

Right lead

0


R
L


Superconductor

Superconductor



)
(
G
)
(
G
d
t
h
e
)
(
I
,
,
LR
,
,
RL











11
11
2


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


is in equilibrium



Even in the perturbed system:



)
(
f
)
(
)
(
)
(
r
a
,




G
G
G






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




11
11
11





)

)


)
(
f
G
G
G
G
d
t
h
e
)
(
I
r
,
LR
r
,
RL
a
,
LR
a
,
RL



11
11
11
11
2






)

)


)
(
f
G
G
G
G
d
t
h
e
)
(
I
r
,
LR
r
,
RL
a
,
LR
a
,
RL



11
11
11
11
2





)
(
f
)
(
D
)
(
g
)
(
g
Im
d
sin
t
h
e
)
(
I
r
,
R
r
,
L







21
12
2
2


1

)
(
D

Tunnel limit










T
k
tanh
sin
eR
)
(
I
B
N
2
2



Ambegaokar
-
Baratoff

]
[
)
(
g
t
)
(
tg
det
)
(
D
r
R
r
L






I
2
2
21
12
)
i
(
)
(
g
)
(
g
r
r











Nambu space

-3
-2
-1
0
1
2
3
0
1
2
)


= 0.1
-3
-2
-1
0
1
2
3
0
1
2
3
4
5
)


= 0.95
  
-3
-2
-1
0
1
2
3
-30
-20
-10
0
10
20
30
j(

)


= 0.95

= 2.5
)
(
f
)
(
D
)
(
g
)
(
g
Im
d
sin
t
h
e
)
(
I







21
12
2
2


0

)
(
D

Andreev states











2
1
2




sin
)
(
)
(
f
)
(
D
)
(
g
)
(
g
Im
d
sin
t
h
e
)
(
I







21
12
2
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/


/









T
k
)
(
tanh
)
(
sen
h
e
)
(
I
B
s
2
2







Supercurrent





d
)
(
d
e
)
(
I
S


Two level system

Josephson supercurrent


2
1
2
2





sen
sen
e
)
(
I
s




0






sen
h
e
I
s


)
(
1


2
2
)
(



sen
h
e
I
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
(

)/I
c

= 0.1
0,0
0,5
1,0
1,5
2,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
I(

)/I
c


=0.9
0,0
0,5
1,0
1,5
2,0
-2
-1
0
1
2
I(

)/I
c


=1
S/S atomic contact with finite bias



Multiple Andreev reflections (MAR)

Sub
-
gap structure
:

qualitative explanation



e

a)


1 quasi
-
particle

eV
>




1
p
e

h

b)

eV
>


2
2


p
e

e

h

c)


3 quasi
-
particles

eV
>
2
3

3
3


p

2 quasi
-
particles

I

V



a



b

3

c

n

quasi
-
particles

eV
>
2

n

Conduction in a superconducting junction


2


2


I

eV

2



F,L

E
F,L
-

E
F,R
= eV > 2


2


E
F,R

I

Experimental
IV

curves in superconducting contacts

0
100
200
300
400
500
0
10
20
30
40
50


T = 17 mK
V [ µV ]
I [ nA ]




3

Al 1 atom

contact

Superconductor

Superconductor

Andreev reflection in a superconducting junction


eV>


I

eV

2




偲潢慢楬i瑹

2


T牡湳浩瑥搠捨慲来g

e
2
Superconductor

Superconductor

Multiple Andreev reflection


eV > 2



I

eV

2




2




偲潢慢楬i瑹

3


T牡湳浩瑥搠捨慲来


e
3
Theoretical model

eV
R
L





eV
dt
d
2


t
eV
t

2
)
(
0




2
/
)
(
2
/
)
(
t
i
R
t
i
L
e
e









2
/
)
t
(
i
te
t


Gauge choice

V



n
t
in
n
e
V
I
t
V
I
)
(
)
(
)
,
(
]
[







L
R
)
t
(
i
R
L
)
t
(
i
R
L
te
te
c
c
c
c
H
H
H








time dependent perturbation

L

R

t

Left lead

Right lead

eV
R
L




Superconductor

Superconductor

dc component of the current I
0
(V)

Calculation of the current

]
[
2
2
)
t
(
c
)
t
(
c
te
)
t
(
c
)
t
(
c
te
ie
)
t
(
I
ˆ
L
R
/
)
t
(
i
R
L
/
)
t
(
i
















)
t
,
t
(
G
te
)
t
,
t
(
G
te
e
)
t
(
I
,
LR
/
)
t
(
i
,
RL
/
)
t
(
i







11
2
11
2
2





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



Non
-
linear and non
-
stationary current

Experiments

]
[







L
R
)
t
(
i
R
L
)
t
(
i
R
L
te
te
c
c
c
c
H
H
H








0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
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





3

0
100
200
300
400
500
0
10
20
30
40
50


T = 17 mK
V [ µV ]
I [ nA ]




3

Al

“one
-
atom” contact

0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
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
V
e


2
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
(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
I
0
(V)


I
0
(V) characteristics

0
1
2
3
0
1
2
3
4
T
1
=0.800, T
2
=0.075
T
1
=0.682, T
2
=0.120, T
3
=0.015
T
1
=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
5
0
2
4
e
d
c
b
a
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
n
h
e
G

2
2
PIN code



n



Correlation between number of channels and number of valence atomic orbitals


3s

3p

Al

eV

7

~



Al

3



Pb

3



Nb

5



Au

1

(Saclay)

(Saclay)

(Leiden)

(Madrid)

MCBJ

MCBJ

MCBJ

STM

Proximity effect

Determination of conduction channels of an atomic contact

Shot noise in superconducting atomic contacts

T
k
eV
B

eI
S
2
)
0
(

Poissonian limit

*
2
/
)
0
(
q
I
S

Charge of the carriers






)
t
(
)
(
)
(
)
t
(
dt
)
(
S
I
I
I
I




0
0
0

0




What is the transmitted charge in a Andreev reflection?



e

eV
>


e

h

eV
>


e

e

h

eV
>
2
3

e
Q

*
e
Q
2
*

e
Q
3
*

?

?

0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
1
2
3
4
5
6
7
8
0.95
Shot Noise
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1.0
S/(4e
2

/h)
eV/



Huge increase of S/2eI for V 0

Theoretical curves

0,5
1,0
1,5
2,0
2,5
3,0
0
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
0.95
q = S/2eI
eV/




Effective charge carried by a multiple Andreev reflection:










eV
Q
2
Integer
1
*
Shot noise measurements in atomic contacts



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

superconducting

Al contact

effective charge


SC

SC

F

S

S

Superconducting transport through a magnetic region

Superconducting transport through a correlated quantum dot