the noise spectra of mesoscopic structures

The noise spectra of mesoscopic structuresEitan Rothstein

With Amnon Aharony and Ora Entin

22.09.10 University of Latvia, Riga, Latvia

The desert in Israel

Outline

• Introduction to mesoscopic physics

• Introduction to noise

• The scattering matrix formalism

• Our results for the noise of a quantum dot

• Summary

Mesoscopic Physics

Meso = Intermidiate, in the middle.

Mesoscopic physics = A mesoscopic system is really like a large molecule, but it is always, at least weakly, coupled to a much larger, essentially infinite, system – via phonos, many body excitation, and so on. (Y. Imry, Introduction to mesoscopic physics)

A naïve definition: Something very small coupled to something very large.

Very high mobilty

GaAs-AlGaAs at the Heiblum group - PRL 103, 236802 (2009)

Si at room temperature

Evdrift

sV

cm

26106

sV

cm

2

1400

Going down in dimensions (2d)

2DEG

Going down in dimensions (1d)

Nanowire and QPCNanowire Quantum point contact

Quantized conductance curve

Going down in dimensions (1d)

Edge states

Under certain conditions, high magnetic fields in a two-dimensional conductor lead to a suppression of both elastic and inelastic backscattering. This, together with the formation of edge states, is used to develop a picture of the integer quantum Hall effect in open multiprobe conductors. M. Buttiker, Phys. Rev. B 38, 9375 (1988).

Going down in dimensions (0d)

Quantum Dots

There are different types of quantum dots.

A large atom connecting to two ledas

A metallic grain on a surface Voltage gates on 2DEG

Going down in dimensions (0d)

Quantum Dots

A theoretical point of view:

Going down in dimensions (0d)

The pictures are taken from the review by L P Kouwenhoven, D G Austing and S Tarucha

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 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:

(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

)0(ˆ)(ˆ)(ˆ ItItIdt RL

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

LJ RJJ JJ J J Jd

ˆ( ) 1/ 2

L L R

d L R R

iS E

E i

2NJ L R

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

Summary

A single level dot

• At and the noise of a 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.

d

( ) ( )C )()( C

0T 0eV

a V

0

Thank you!!!

“The noise is the signal” R. Landauer, Nature London 392, 658 1998.