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