ELECTRICAL TRANSPORT REGIMES

+ +++

+++++ le

φL

�

�

diffusive

ballistice

e

l L

L l

Inelastic scatteringCoulomb interactions

PhononsMagnetic moments

φLElastic scatteringGrain boundaries

EdgesImpurities

el

λλλλ

�

�

�

�

, , 3D

2

, 1

, , 0

F x y z

F z

F y z

F X y z

L L L

L D

L L D

L L L D

φ

φ

�

�

fully coherent

incoherent

L

L

L

Lconfinement

EXPERIMENTAL PLAYGROUND

80 (nm) 0.5

50meV 10eV

15�m 30nm

10 (�m) 101K

F

F

e

E

l

LT

φ

λ

<

( )3 210 mµ −Low density metals High density metals ( )10 310 mµ −

Epitaxial heterostructures Evaporated thin films

M. Heiblum et al, Weizmann Inst., IsraelN. Birge et F. Pierre, MSU, USA

TRANSPORT AS A SCATTERING PROBLEM

V

Fully Coherent Conductor

Landauer (1957)Büttiker (1992)

Iφ<<L L

Ideal lead

Lµε

1

)(εLf

Reservoir

Ve

ε

Rµ

1

)(εRf

Reservoir

Ideal lead

µ µ− =L R eV

SCATTERING MATRIX

( ) ( )( ) ( )

LL LR

RL RR

S S

S SS

� �� �=� �� �

L L

RR

b a

abS

� � � �=� � � �� � � �� �

� �

� �

( ),1

( )

( ),

L R

L R

L R N

a

a

a

� �� �= � �� �� �

��

,1

,

L

L

L N

b

b

b

� �� �= � �� �� �

��

Incoming modes

Outgoing modes

† † 1SS S S= =Current conservation

unitary

,1La

,2La

,3La

,1Ra

,2Ra

,3Ra

,1Lb

,2Lb

,3Lb

,1Rb

,2Rb

,3Rb

INDEPENDENT CONDUCTION CHANNELS

( ) ( ) ( )

( ) ( ) ( )

L R L R L R

L R L R L R

a U a

b V b

′ =

′ =

� �

� �

Unitary transformation

( ) ( )( ) ( )

† 1/ 2 1/ 2

† 1/ 2 1/ 2

0 00 0

LL LRL L

R RRL RR

S SV U iR TS

V U T iRS S

� �� � � � � �−� �′ = =� � � � � �� � −� �� � � �� �

With R and T real, diagonal, positive matrices

Collection of independent 1D problems

L L

R R

b aS

b a

� � � �′ ′� � � �′=� � � �′ ′� � � �

� �

� �

Th. Martin and R. Landauer, PRB 45, 1742 (92)

IDEAL 1D LEADS

2 2( ) / 2k k mε ± = �1

exp ( / )( )

εε

− �Li k x thv

If all states in energy range δεare occupied the current is

( )εδ δε δεε

= =dndL

v eI e

h

density of states in 1D

( )dn Ld hvε ε

=( )electron velocity

/for F F

kv eV

v ε εε

= ∂ ∂≈

�

�

Maximum rate of fermions emitted by a source into a 1D lead:1 ε=�dN dh

Rk

)( kε

δε

SINGLE CHANNEL SCATTERING MATRIX

( ) '( )( )

( ) '( )r t

St r

ε εε

ε ε� �

= � �� �

† † 1SS S S= =Current conservation

2 2 2 2

2 2 2 2

1

1

′ ′+ = + =

′ ′+ = + =

r t r t

r t r t

0

0

t r r t

t r r t

∗ ∗

∗ ∗

′ ′+ =′ ′+ =

2 2

2 2

transmission probability

= 1 reflection probability

t t

r r

τ

ρ τ

′= = =

′= = − =

r r’

t

t’

EXAMPLE: DELTA BARRIER

2 2

02 ( )2

ψ δ ψ εψ∂− + =∂

�U x

m x

( ) ( )

02

02

02

2( 0) ( 0) (0)

2

is continuous at 0

Derivative discontinuity t

a 0,

2L L R R R R

L L R R

L L R R R R

mU

mUika ikb ika ikb a b

a b a b

mUa b a

x

a

x

b b

ψ ψ ψ

ψ

′ ′+ − − =

� − + − + = +

� + = +

� − + − +

=

=

=

�

� �

( )0

0 0

0

0 0

22 2

22 2

iUiU iU

SiU

iU iU

εε ε

εε

ε ε

−� �� �+ +� �=� �−� �+ +� �

22

2mk

ε=�

ikx ikx ikx ikxL L R Ra e b e a e b e− −+ +

0 x

U0

Schrödinger eq.

SCATTERING STATES( ) ( )

( )

left

right

( )(e )1

)( , )

(

L L

R

ik x ik x

ik xF

L

e

vt

he

r

x

ε ε

εεε

εϕ

−� += �

( ) ( )

( )

right

left

( )((

( )

e )1

, )

R R

L

ik x ik x

ik xF

R

r

t

e

xhv

e

ε ε

ε

εε

ϕε

−

−

� + ′=

′

�

Complete orthonormal basis of current carrying states

rt

Li k xe

τ δεδ =L

eh

I

τ δεδ =R

eh

I

Ri k xe−

r’

t’

FILLING OF THE SCATTERING STATES BY THE RESERVOIRS

Lϕ Rϕ

r r’

t

t’Rk

)( kε

Lµ

Lk

)( kε

RµVe

Transport described using eigenstates in equilibrium with different reservoirs

[ ] [ ]

[ ]

( ) ( )( ) ( ) ( )

(( )

( )

) ) )( (

δ τ εε δ ε

ε

ε δ δε

ε τ ε

ε

ε

= − = −

= −�

RL L L

L

R R

R

f I ff I f

f

eI V

h

eV d fI

h

CONDUCTANCE from TRANSMISSION

[ ]( ) (( ))( )L Rfe

I V dh

fε εε τ ε= −�

2

0

( )lim ( )V

I e fG d

V hεε τ εε→

∂= =∂�

Current

Conductance

LANDAUER FORMULA

MULTICHANNEL2

ii

eG

hτ=

2

τ= eG

h2 (spin)×

2

0 38�S 1/ 26k�= � �e

Gh

If over the energy range between µL and µR( )τ ε τ=

[ ] ( )2

((( ) )) ετ ε τµτε µ= − = − =� L LR R

e e eI V d V

h h hf f

TRANSPORT from TRANSMISSION

{ττττ1111,..., ,..., ,..., ,..., ττττΝΝΝΝ}Normal state Superconducting

state

Conductance

Shot Noise

Josephsonsupercurrent

Shot NoiseCoulomb Blockade

…

(‘86)

(‘91)

(‘99)

(‘81)

(‘89)

(‘95)

(‘01)(‘01) Shapiro Steps

Current-Voltagecharacteristic

…

Thermopower

MESOSCOPIC PIN CODE

CONDUCTANCE QUANTIZATION

ττττ�=1 for all open channels

0<ττττN<1 for the last channel

( )2 22 2

1i N

e eG N

h hτ τ= = − +

van Wees; Wharam (1988)

T < 1 K

Counter-support

Insulating layer

Pushing rod

QUANTUM POINT CONTACTS IN METALS

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

��������������

�� ����������

0.0 0.1 0.2 compression (nm)

tunnel regime

Aluminum T = 50 mK

N state (H=0.2 T):

�� ��

�

-4 -2 0 2 4

-3

-2

-1

0

1

2

3 ∆ = 178 µVIc = 22.9 nA

I / I c

VJ / (∆ / e)

DC Josephsonsupercurrent peak

Multiple AndreevReflections

Counter-support

Insulating layer

Pushing rod

SUPERCONDUCTING ATOMIC CONTACTS

S state (H=0):

SUPERCONDUCTING RESERVOIRS

Ideal leadIdeal leads s

• New quasiparticles

• New scattering mechanism

NORMAL REFLECTION

)( kε

)( kε

An incident up-spin electron is reflectedas an up-spin electron of the same energy

ANDREEV REFLECTION

)( kε)( kε

S

An incident up-spin electron with energy µ+ε« drags » with it

a down-spin electron of energy µ−εto form a Cooper pair in the superconductor

leaving behind a hole in the spin-down band

Pairingpotential

ie φ∆ = ∆

COUPLING OF e���� AND h����

0 /Fvξ = ∆�

0ξ

S

0ξ

SN

N-S interface

Bulk S

BOGOLIUBOV-de GENNES EQUATION

Spin-up electrons ∂ =∂� N

ui H u

t

Spin-down holes ∗∂ = −∂� N

vi H v

t

Conjugated particles

† † †, ( ( ) )h e h ec cψ ψ ∗= =

( )N

N

H u ui

H v vt

µ

µ∗ ∗

− ∆� �� � � �∂=� �� � � �� �∆ − − ∂� � � �� ��

( ) /( )( )E

ik E x iEtu Ee e

v Eψ ± ± −� �

= � �� �

�

Propagating solutions of BdG equation

SUPERCONDUCTING QUASIPARTICLE STATESelectron-like and hole-like

( )

( )

2( ) 2

( )

22( )

1sgn( )

12 sgn( )

e h

ie h

e h

uE E E

v e

k m E E

φ

µ

= ± − ∆∆

= ± − ∆�

for or 0

( ) ( )1 0and

( ) ( )0 1e h

e h

E

u E u E

v E v E

∆ ∆ →

� � � �� � � �→ →� � � �� � � �

� � � �� � � �

�

-3 -2 -1 0 1 2 30

1

2

3

4

E ∆

( )( )n F

n E

n ξ ε=

Density of states

evanescent

ANDREEV REFLECTION AMPLITUDEie φ∆

x

( )1 0,

0 1e

e

ua E

vφ

� �� � � �+ = � �� � � �

� � � � � �

( ) ( )( )221, sgne

ie

va E E E E

u e φφ = = − − ∆∆

Matching at x=0

-3 -2 -1 0 1 2 3

2φ π+

φ π+

φ

E ∆-3 -2 -1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

E ∆

( ),a E φ ( )( ),Arg a E φ

ALL TOGETHER NOW…

Normal + Andreev scattering

s s

s sRegion I

(N)Region II

(N)

MULTIPLE ANDREEV REFLECTIONS (MAR)

2E neV+

( )2 1E n eV+ +

E

E eV+

2E eV+

3E eV+

MAR + NORMAL SCATTERING

2E neV+

( )2 1E n eV+ +

( )2 1E n eV+ −

E

Scatterer

Region I(N)

Region II(N)

Superconductor (L)Superconductor (R)

SCATTERING STATES

2E neV+

( )2 1E n eV+ +

( )2 1E n eV+ −

nB2n na A

nA2n na B

1nD −

2 1 1n na C− −

1nC −

nC

2 1n na D+

nD

( )= +na a E neV

2 0

2 1

n n n n

n n n

B a A JS

C a D

δ

+

+� � � �=� � � �

� � � �

2

2 1 1

n n n

n n n

A a BS

D a C∗

+ −

� � � �=� � � �

� � � �

electrons

holes

1/ 22( ) 1 ( )� �= −� �

J E a E

a.c. JOSEPHSON CURRENTS

( ) ( )( ) ( )2 /1 2 0 2

N N N Ne e h h i E neV tik x ik x ik x ik x

n n n n n n nn

x a A J E e B e A e a B e eψ δ − +− −

∈

� �= + + + +� � �

( ) ( )( )2 1 /2 2 1 2 1

N N N Ne e h h i E n eV tik x ik x ik x ik x

n n n n n nn

x C e a D e a C e D e eψ − + +− −+ +

∈

� �= + + +� �

�

0 2eVω = �

( )( )( )( )

2 2 2

02 2( )

2tanh

12δ

∗− −

∗ ∗ ∗+ + +

� �� �+� � � �= − �� �� �+ + −� � � �� �� �

�

k k k k

k kn n k n n k n n kB

n

J E a A a Ae EI eV dE

a a A A B Bh k T

( ) ( ) ( )0 0, ik tk

k

I V t I V e δ ω+∞

+

=−∞

= Currents at all harmonics of

Josephson frequency

dc MAR CURRENT

( ) ( ) ( )( )2

2 2 20 0 0 0 2

2tanh 1

2 n n nnB

e EI eV dE J E a A A a A B

h k T∗� �� �� � = − + + + − �� �� �

� � � �� � �

� � � � ��

�

�

�

�

���τ ��� ���

���

���

������������∆��

�����∆��� ����� ! �"�#$%���"����������&�'(�� ���)� *�"�#$����"���++�����+����,�(���-�)� *�"�%.#���"�����������

MULTIPLE CHARGE TRANSFER PROCESSES

���

��

Blonder, Tinkham, Klapwijk (‘82)

�∆ ����∆ ����∆ ���

��

τ

S S

SUPERCONDUCTING ATOMIC CONTACTS IV

0 1 2 3 40

1

2

3

����������∆��

eV / ∆

τ3=0.09

τ2=0.11

τ1=0.55

exp 0 ( , )n

nI I V τ=

Number of channels fixed by number of valence orbitals

Fingerprint ofmesoscopic PIN code

0

5

10

15-60 -40 -20 0 20 40 60

{ 0.21, 0.20, 0.20 } { 0.40, 0.27, 0.03 } { 0.68, 0.25, 0.22 } { 0.996, 0.26 } Theory Poisson's Limit

�,�������/���������01��

�� ( nA )

-20 -10 0 10 20

������������/���������01��

Shot Noise in the Normal State

�

τ

�

Spectral density SI

Single channel ,� ��������/τ �

Multi-channel ,� ����������/ Στ�� ��Στ� �

Partition Noise {τι} known from the superconducting fingerprint

1 2 3 410-4

10-3

10-2

��2�����"�����"������3

��������

������∆

�,����������01��

SHOT-NOISE IN SUPERCONDUCTING STATE

�� � �

1 2 3 40

1

2

3

4

5

������∆

�45����

�����2�����"�����"������3

Effective Charge Q* = SI / 2eI

����6�$�.�� 7

$��' �"�6��8���99� �"�:��;�)��� �!�'��< =�� "�#$%��+"������������>��'��'(�� ���)� *������?��@ �����:����� ��������6�$�.�� 7

0 1 2 3 4 50.1

1

10 �2�����+"����&��3

�2�����"���&&"�����"������3���

�2���+�"����&������3

��6�$�.�� 7

Q *

/ e

eV / ∆

LARGE CHARGE QUANTA

SUMMARY

Scattering theory provides a rigorous framework to understand transport phenomena

Simple extension of equilibrium statistical mechanics:

( )

( )

ˆ

and eigenstates and eigenenergies of system

ˆ ,

scattering states

ε

ε

ε µ

= −

= −

nn

n

s

F

sss

A n A n f E

n

A s A s f

s

T

At equilibrium

Each scattering state at equilibrium with a different reservoir

Out of equilibrium

In superconducting mesoscopic systems the BdG equation can be used to construct scattering states in much the same way that the Schrödinger equation is used

in normal mesoscopic systems

� S. Datta : Electronic transport in Mesoscopic systems, Cambridge University Press(1997).

� C. Glattli, in « Quantum Entanglement and information processing », Les Houches Lecture, Session LXXIX, Elsevier (2004).

� S. Datta, P. Bagwell, and M.P. Anatram, « Scattering Theory of Transport for Mesoscopic Superconductors », Phys. Low-Dim. Struct., 3 pp1-58 (1996).

� M. Büttiker, Phys. Rev. B, 46 12485 (1992).

� Th. Martin and R. Landauer, Phys. Rev. B, 45 1742 (1992).

� G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B, 25 4515 (1982).

� C. Beenakker and H. van Houten, Quantum Transport in SemiconductorNanostructures, in `̀Solid State Physics'', Ehrenreich and Turnbull Eds., Academic Press(1991).

BIBLIOGRAPHY