Quantum interference

� Aharonov-Bohm effect� ”Aharonov-Bohm” ring� Basic properties of

superconductors� Andreev reflection� Andreev conductance

� Multi-terminal conductors� Scattering matrix� Multi-terminal Landauer formula� Example: Beam splitter� Quantum interference� Example: Double barrier

Multi-terminal conductors

4

5

1I

3I

2I1V

2V

2V

We want to express currentsin terms of voltages

Scattering region

Reservoir

Ideal lead

Gate

Linear regime

4

5

1I

3I

2I1V

2V

2Vk kl l

lI G V=∑

klG - conductance matrix

0 0k klk k

I G= ⇒ =∑ ∑Current conservation:

No current if the potentials at all leads are shifted by an equal amount

0kll

G =∑

Two-terminal conductors

1 1

2 2

I VG GI VG G

− = −

1V2V

1I 2IConductance matrix: Only one independent element

Scattering matrix

;m nsβ α

-amplitude of propagation from the terminal α,transport channel n to the terminal β, transportchannel m....

...

...

1N1 scattering region

s11,12s12,12s13,12

s 21,12

s 22,12

s 23,12

s31,12

s32,12

s33,12

Properties of scattering matrix

;n msα α - reflection back from α.

; ,n msα β α β≠ - transmission from β to α.

Unitarity: �� � 1s s =*

; ;n l n m lmn

s sα γ α β βγα

δ δ=∑Expresses conservation of the

number of particles

Symmetry with respect to the time reversal:

; ;( ) ( )n m m ns B s Bα β β α= −

Landauer formula

( / ) ( )QI G e dE P f Eα αβ ββ

= − ∑∫Relates currents to voltages by means of the scattering matrix

(trace is taken over transport channels)

Current conservation: follows from unitarityNo current at equilibrium (all voltages are the same)-

also follows from unitarity

��P Tr s sαβ αβ αβ αβδ = − Probability of transmission from α to β:

Fool-proof check:

Landauer formula

��QG G Tr s sαβ αβ αβ αβδ = − − Linear regime:

Relation to the two-terminal formula: α,β=L,R

( )� ��LR Q LR LR QG G Tr s s G Tr t t = =

Time-reversal symmetry: ( ) ( )G B G Bαβ βα= −

(In accordance with Onsager symmetry relations)

Example: Beam splitter

3

1

2

a new scattering element

1/ 2 1/ 2 1/ 2

� 1/ 2 1/ 2 1/ 2

1/ 2 1/ 2 0Bs

−

= −

(will use later for quantum interference!)

Example: Beam splitter

3

1

2

Conductance matrix:

3/ 4 1/ 4 1/ 21/ 4 3/ 4 1/ 21/ 2 1/ 2 1

QG Gαβ

− = − −

Example:

1 2 3; 0V V V V= = =QG V

/ 2QG V

/ 2QG V

Quantum interference

Repeat basic quantum mechanics: Double-slit experiment

A B

1

2

Probability of propagation from A to B:

21 2

2 2 *1 2 1 22 Re[ ]

ABP A A

A A A A

= +

= + +

1P 2P

Classical

Interference termQ-mechanical!

Two barriers

L R

Try to guess the result:

Resistances are added - Classically

Is quantum interference important?

1

totr

tott

Need to learn how to combine scattering matrices.

Two barriers

L R

1

totr

totta

b

c

d

'L L

L L

r tt r '

R R

R R

r tt r

00

i

i

ee

ϕ

ϕ

Scattering matrix of the left barrier

Scattering matrix of the right barrier

Propagation between the barriers

kdϕ =

Two barriers

L R

1

totr

totta

ide ϕ

iae ϕ

d

'1ot

iL

L

t L

L

ra

r tr dt e ϕ

=

' 0tot

Ri

R

R R

d aetrt

rt

ϕ =

Two barriers

L R

1

totr

tott

2( )1 2 cos

L Rtot

L R L R

T TT E tR R R R χ

= =+ −

χ � phase accumulated during the round-trip

21 'L R

tot iL R

t ttr r e ϕ=

−

Why does it have anything to do with the double-slit experiment?

Two barriers

21 'L R

tot iL R

t ttr r e ϕ=

−

Process Amplitude Probability

iL Rt t e ϕ

L RT T

3' iL R L Rt t r r e ϕ

L R L RT T R R� � �

Sum of amplitudes:

Sum of probabilities: 1L R

clL R

T TTR R

=−

Two barriers

( )1 2 cos

L R

L R L R

T TT ER R R R χ

=+ −

Quantum result: transmission coefficient depends on energy(Classical result: It does not)

( )T E

E

Take 1L RT T= !

Valley: 2( ) LT E T∝Peak: max 1 (@ 0)T χ= =

Resonant tunneling

ddE

χ τ=

τ � time for the round-trip

Aharonov-Bohm effect

Φ

1

2

Two trajectories enclosing magnetic flux

Phase: ikxWith the vector potential:

( )ek k A xc

→ −" " "

#

1,2 1,21,2

ekL A dlc

θ = − ⋅∫""

#

Aharonov-Bohm effect

Φ

1

2

2 1 2 1( ) ek L L A dlc

θ θ− = − − ⋅∫""

# $

0

2e eA dlc c

πΦ⋅ = Φ =Φ∫

""# #$

02 c

eπΦ = # - flux quantum

All quantities are periodic in Φ, even if there is no magneticfield at the trajectories.

“Aharonov-Bohm” ring

1 tΦ1 2

1b

1d

1a

1c

2b2a

2c2d

0

/ 22 /kLχ

ϕ π== Φ Φ

r

0 1/ 2 1/ 2

1/ 2 1/ 2 1/ 2

1/ 2 1/ 2 1/ 2

− −

1/ 2 1/ 2 1/ 2

1/ 2 1/ 2 1/ 2

1/ 2 1/ 2 0

−

−

/ 2

/ 2

00

i i

i i

ee

χ ϕ

χ ϕ

+

−

/ 2

/ 2

00

i i

i i

ee

χ ϕ

χ ϕ

−

+

Left beam splitter

Lower arm

Right beam splitter

Upper arm

“Aharonov-Bohm” ring

1 tΦ1 2

1b

1d

1a

1c

2b2a

2c2d

1 1

1 1

0 1/ 2 1/ 2 11/ 2 1/ 2 1/ 2

1/ 2 1/ 2 1/ 2

rb ad c

= − −

0

/ 22 /kLχ

ϕ π== Φ Φ

r

“Aharonov-Bohm” ring

1 tΦ

2

2 2

2sin (1 cos )sin 2 [cos2 (1/ 2)(1 cos )]

T χ ϕχ χ ϕ

+=+ − +

02 /ϕ π= Φ Φ

2

(1 cos )2[3 cos ]2

T ϕϕ

πχ +→ =+

= Flux dependent!

“Aharonov-Bohm” ring

02 /ϕ π= Φ Φ2

(1 cos )2[3 cos ]2

T ϕϕ

πχ +→ =+

=

“Aharonov-Bohm” ring

1 t

Φ 02 /ϕ π= Φ Φ

2 20 1 1 2 2

i i i it e e e eϕ ϕ ϕ ϕα α α α α− −− −= + + + + +…

Classical

One turn clockwiseOne turn

counterclockwise

Two turns