The noise spectra of mesoscopic structuresEitan Rothstein

With Amnon Aharony and Ora Entin

02.02.09 Condensed matter seminar, BGU

Outline

• Classical vs. quantum noise

• The noise spectrum

• The scattering matrix formalism

• A single level dot

• Two single level dots

• Summary

Classical Noise

The Schottky effect (1918) 2S e I

Discreteness of charge

Classical Noise

Thermal fluctuations

Nyquist Johnson noise (1928) TGkS B4

Quantum Noise

Quantum statistics

M. Henny et al., Science 284, 296 (1999).

Quantum Noise

Quantum interference

I. Neder et al., Phys. Rev. Lett. 98, 036803 (2007).

The noise spectrum

' 'ˆ ˆ( ) ( ) (0)

i tC dte I t I

ˆ ˆ ˆI I I ,L R

' ,L R

' '

*( ) ( )C C

L R

... - Quantum statistical average

Sample

Different CorrelationsNet current:

Net charge on the sample:

Cross correlation:

Auto correlation:

)ˆˆ(2

1ˆRL III

)ˆˆ(2

1ˆRL III

))()()()((4

1)()( RLLRRRLL CCCCC

))()()()((4

1)()( RLLRRRLL CCCCC

))()((2

1)()( RLLR CCC

( ) 1( ) ( ( ) ( ))

2auto

LL RRC C C

Relations at zero frequency

)(ˆ)(ˆ)(ˆ

tItIdt

tnde RL

ˆ( ) ˆ (0)d n t

e dt Idt

Charge conservation:

ˆ ˆ ˆ( ( ) ( )) (0)L Re dt I t I t I

(0) (0)L RC C

ˆ ˆˆ ˆlim ( ) (0) ( ) (0)e n I n I

0

*' '( ) ( )C C (0) (0) (0) (0)LL RR RL LRC C C C

( ) 1(0) ( (0) (0) (0) (0)) 0

4 LL RR LR RLC C C C C

( ) 1(0) ( (0) (0) (0) (0)) (0)

4 LL RR LR RL LLC C C C C C

The scattering matrix formalism

M. Buttiker, Phys. Rev. B. 46, 12485 (1992).

1/)( ]1[)( TkE BeEf

Analytical and exact calculations

No interactionsSingle electron picture

( ) ( )( )

( ) ( )LL LR

RL RR

S E S ES E

S E S E

( )

( )

2( )

( )

( )

( , ) ( )(1 ( ))

( , ) ( )(1 ( ))

( )8

( , ) ( )(1 ( ))

( , ) ( )(1 ( ))

LL L L

LR L R

RL R L

RR R R

dEF E f E f E

dEF E f E f Ee

C

dEF E f E f E

dEF E f E f E

2**)( )()()()(1),( ESESESESEF RLRLLLLLLL

2**)( )()()()(),( ESESESESEF RRRLLRLLLR

The scattering matrix formalism

RLLRRRLL CCCCC

4

1)()(

' 'ˆ ˆ( ) ( ) (0)

i tC dte I t I

2

'' '

' ,

( , ) ( ) 1 ( )2 L R

eF E f E f E

A single level dot

LJRJJ JJ J J Jd

ˆ( ) 1/ 2

L L R

d L R R

iS E

E i

2NJ L R

E. A. Rothstein, O. Entin-Wohlman, A. Aharony, PRB (in press).

Unbiased dot

d

L R

0TkB3TkB5TkB

• Resonance around

• Without bias, is independent of

• , parabolic around

d

LR

LRa

)()( C

0)0()( C 0

a

(In units of )

d

Unbiased dot0a7.0a1a

LR

LRa

0TkB

aa

[ ] 4Bk T

• At maximal asymmetry (the red line), , and

• Without bias the system is symmetric to the change

0)()( C )()( )()( CC

0• The dip in the cross correlations has increased, and moved to • Small dip around ( ) ( )dC

A biased dot at zero temperature

LR

LRa

7.0a0a7.0a1a

1a

• , parabolic around

• When , there are 2 steps .

• When , there are 4 steps .

• For the noise is sensitive to the sign of

( ) (0) 0C 0

| | 2 | |deV

2 deV 2 deV

2 deV

0

| | 2 | |deV d

/ 2L eV / 2R eV

a

A biased dot at zero temperature

LR

LRa

• The main difference is around zero frequency.

2 deV 2 deV

2 deV

7.0a0a7.0a1a

1a

A biased dot at finite temperature

LR

LRa

• For , the peak around has turned into a dip due to the ‘RR’ process.

• The noise is not symmetric to the sign change of also for

0.7a 0

a 0

[ ] 22eV [ ] 3Bk T

7.0a0a7.0a1a

1a

Two single level dots

1 21 2 2 1

ˆ( , ) 1 ( , ) ( ) ( ) ( , )4

S E ig E E E i D E

,L R

2 21 2 2 2 1 1( , ) ( , ) ( ) ( )

i i

LR L R L RS E ig E e E e E

*( , ) ( , )RL LRS E S E

1

1 21 2( , ) ( , )

2 2

i ig E E E D E

1 2 1 2 1 2 1 2

1 1( , )

4 2L L R R L L R RD E Cos

Unbiased dots 0a

1a

1 1

1 1

R L

R L

a

1

L R

20TkB

•Each resonance has one step

7.0a

12

Unbiased dot0a7.0a1a

LR

LRa

0TkB

Unbiased dots 0a

0.5a 1a

1 1

1 1

R L

R L

a

0TkB

• There is a dip at

• The dip in is a function of

L 1 R

2

1 2 ( ) ( )C a

Finite temperature

0a 0.5a 1a

1 1

1 1

R L

R L

a

4Bk T

• There is a dip at for both cases.1 2

1

L R

2

L 1 R

2

new

AB flux 1 1

1 1

0R L

R L

a

0Bk T

• The dip in oscillates with AB flux.

L 1 R

2( ) ( )C

Biased dots 1 1

1 1

R L

R L

a

0TkB

•If there is a dip/peak at 1 2

1

L

R

2

1

LR

2

1

L R

2

1 2,R

7.0a0a7.0a1a

1a

Summary

A single level dot

• At and the single level quantum dot exhibits a step around .

• Finite bias can split this step into 2 or 4 steps, depending on and .

• When there are 4 steps, a peak [dip] appears around for [ ].

• Finite temperature smears the steps, but can turn the previous peak into a dip.

2 single level dots

• If , there is a dip / peak at .

• This dip oscillates with .

d

( ) ( )C )()( C

0T 0eV

a V

0

1 2,i 1 2

Thank you!!!