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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
tLeft lead Righ t lead
eVRL
SuperconductorNormal metal
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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
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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
tLeft lead Righ t lead
0 RL
SuperconductorSuperconductor
Superconduct ing phase dif ference
RLLi
L e RiR e
)(t LRRLRL
ccccHHH
BCS sup ercondu ctors
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12
SQUID configuration
transmission
L
Li
L e
L
L
i
i
L
a
LLe
e)(i)(
22
1g
Nambu space
Uncoup led superconductors
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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
t
Left lead Righ t lead
0 RL
SuperconductorSuperconductor
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)(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
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) ) )(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
)()(
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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
21122
2
0)(DAndreev states
2
1 2 sin)(
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)(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
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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
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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
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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
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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
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Superconductor
Superconductor
Andreev ref lect ion in a superconduct ing jun ct ion
eV>
I
eV2
Probability
2
Transmitted charge e2
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Th ti l d l
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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
tLeft lead Righ t lead
eVRL
SuperconductorSuperconductor
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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
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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
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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
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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/
Fit t ing of the cu rves I0(V)
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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
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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
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The PIN code of an atom ic contact
n
nh
eG
22
PIN coden
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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
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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* ? ?
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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
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
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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/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*
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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
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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