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The noise spectra of mesoscopic structures. Eitan 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 - PowerPoint PPT Presentation
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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.
