superconducting transport superconducting model hamiltonians: nambu formalism current through a...
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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
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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
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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)
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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
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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
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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
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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
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A word on notation: Nambu space + Keldish space
Superconductivity Non-equilibrium
)'t,t(G ,j,i ,,
21,j,i
Keldish
Nambu
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N/S superconducting contact
Single-channel model
)(t LRRLRL
ccccHHH perturbation
L R
tLeft lead Right lead
eVRL Superconductor
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Superconducting right lead (uncoupled):
R
22
1)(i)( R
aRRg
)(f)()()( RrRR
aRR
,RR ggg
0R
Nambu space
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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
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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
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-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
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Normal metal Superconductor
Andreev Reflection
Probability 2Transmitted charge e2
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)(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
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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
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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
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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
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12
SQUID configuration
transmission
L
LiL e
L
L
i
i
LaLL
e
e)(i)(
22
1g
Nambu space
Uncoupled superconductors
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)(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
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)(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
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)(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
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-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)(
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)(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
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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
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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
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Conduction in a superconducting junction
2 2
I
eV2
EF,L
EF,L - EF,R = eV > 2
2EF,R
I
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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
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Superconductor
Superconductor
Andreev reflection in a superconducting junction
eV>
I
eV2
Probability 2Transmitted charge e2
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Superconductor
Superconductor
Multiple Andreev reflection
eV > 2/3
I
eV22 /3
Probability 3Transmitted charge e3
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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
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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
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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
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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
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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)
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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
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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
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The PIN code of an atomic contact
n
nh
eG
22PIN code n
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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
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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* ? ?
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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
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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*
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Shot noise measurements in atomic contacts
• Cron, Goffman, Esteve and Urbina, Phys.Rev.Lett. 86, 4104, (2001).
superconducting Al contact
effective charge
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SC SC
FS S
Superconducting transport through a magnetic region
Superconducting transport through a correlated quantum dot