The noise spectra of mesoscopic structures Eitan Rothstein With Amnon Aharony and Ora Entin 22.09.10...
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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
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The desert in Israel
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Outline
• Introduction to mesoscopic physics
• Introduction to noise
• The scattering matrix formalism
• Our results for the noise of a quantum dot
• Summary
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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.
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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
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Going down in dimensions (1d)
Nanowire and QPCNanowire Quantum point contact
Quantized conductance curve
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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).
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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
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Going down in dimensions (0d)
Quantum Dots
A theoretical point of view:
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Going down in dimensions (0d)
The pictures are taken from the review by L P Kouwenhoven, D G Austing and S Tarucha
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Classical Noise
The Schottky effect (1918) 2S e I
Discreteness of charge
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Classical Noise
Thermal fluctuations
Nyquist Johnson noise (1928) TGkS B4
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Quantum Noise
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Quantum Noise
Quantum statistics
M. Henny et al., Science 284, 296 (1999).
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Quantum Noise
Quantum interference
I. Neder et al., Phys. Rev. Lett. 98, 036803 (2007).
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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
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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
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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
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The scattering matrix formalism
M. Buttiker, Phys. Rev. B. 46, 12485 (1992).
1/)( ]1[)( TkE BeEf
Analytical and exact calculations
No interactionsSingle electron picture
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( ) ( )( )
( ) ( )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
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LJ RJJ JJ J J Jd
ˆ( ) 1/ 2
L L R
d L R R
iS E
E i
2NJ L R
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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
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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
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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
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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
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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
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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.