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The noise spectrum of a quantum dot
Eitan Rothstein
Netanel Gabdank
Ora Entin-Wohlman*
Amnon Aharony*
Physics Department, Ben Gurion University of the Negev
* Also at Tel Aviv University
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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Mesoscopic Physics
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Quantum dot
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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The model
The model
Quantum-Dot (QD)
N∆
n
Vn , k
k
…
Left (L)
electronic
reservoir
Vn , p
∆
Right (R)
electronic
reservoir
p
-N∆
Vg
DC conductance
Noise spectrum
C ( )   dteit Iˆ(t )Iˆ(0)
I  GV
 I  ( IL  IR ) / 2

V   L   R
G 2 / e 2
Iˆ  Iˆ  Iˆ

 Iˆ  ( IˆL  IˆR ) / 2
F / 
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2C ( ) / e2
No interaction
Hartree Fock
/
The model
The Hamiltonian
N
E
Hˆ    cˆ cˆ Vn,k (cˆ dˆn  h.c)    n dˆ dˆn  C (  dˆn† dˆn  N g ) 2
2 n
n N
 N









†
† ˆ
   p cˆ p cˆ p  Vn, p (cˆ p d n  h.c)
electronic
Hˆ QD charging energy
†
k k k
N
†
k
†
n
states on QD
momentum states in
R/L reservoirs
coupling
R/L reservoir-QD
Hartree Fock approximation
Hˆ effQD 
dˆn†dˆn dˆm† dˆm  dˆn†dˆn dˆm† dˆm  dˆm† dˆm dˆn†dˆn  dˆn†dˆm dˆm† dˆn  dˆm† dˆn dˆn†dˆm
N
ˆ †dˆ  E ( dˆ †dˆ  N )dˆ † dˆ  E
ˆ † dˆ dˆ †dˆ  [const .]

d
d
 n n n C nm n n
m m
C
m n
n m
n N
n m

 


Hartree
Fock
Self-consistent calculation
F
d n†d m   

dE

QD
1
Im[ E  H eff
 i]nm
 N nm
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 /   (L  R ) / 
F / 
Level Occupancy and Effective Hamiltonian
The effective Hamiltonian
Hˆ effQD 
N
 ~ dˆ dˆ  V
n  N
†
n n n
n m
~ ˆ† ˆ
mn d n d m
~n   n  EC ( N mm  N nn  N g )

m
~

Vnm   EC N mn
Level Occupancy
N nn
Effective single-e energies
~n
 F  3.5
 F  3.5
(Hartree)
(Hartree)
EC
N nn ; N nm
* energy
scaled
with 
EC
~
~n ;Vnm
 F  3.5
(Hartree Fock)
 F  3.5
(Hartree Fock)
Hartree terms
Hartree terms
Fock terms
Fock terms
EC
EC
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The Scattering Formalism
Treating the problem in the Landauer Büttiker formalism
Tight-Binding model
Current and correlations
 ( r  L, t )
scattering
states
QD 1
i 2nm [ E  H eff
]nm  L

S ( E )  1 
QD 1 
1  i nm [ E  H eff ]nm  R L
Iˆ (t ) Iˆ (t ' ) 
e2
(2 )2
Iˆ (t )Iˆ (t ' ) 
d 4E
  (2 )2
e2
(2 )2
L R 

R 
e
Iˆ (t ) 
2

S(E)
 ( r  R , t )
scattering
region
scattering
states
dEdE '
A  ( , E , E ' )aˆ † aˆ ei ( E  E ') t

2 
*
A ( , E, E ' )    S
( E )S ( E ' )
e
Ιˆ 
dE | S ( E ) |2 ( f  ( E )  f ( E ))

2
 A ( , E, E ' )A (  , E ' ' , E ' ' ' )aˆ ( E )aˆ ( E ' )aˆ ( E ' ' )aˆ ( E ' ' ' )e

†
†
A ( , E , E ' )A (  , E ' , E )  f ( E )(1  f ( E ' )) e
 d E 

2
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i ( E  E ')( t t ')
(Landauer)
i ( E  E ' ) t i ( E ''  E ' '' ) t '
e
(Büttiker)
The noise spectrum
Noise spectrum
C ( )   dteit Iˆ(t )Iˆ(0)
Calculations performed for kBT  0 and L  R   F , so the noise given by
F
e2
C ( ) 
4
dE[2  r * ( E )r ( E   )  2t * ( E )t ( E   )  r '*( E )r ' ( E   )]

 
F

2C ( ) / e2
2C ( ) / e2
symmetric
L  R
symmetric
 F  0.5
L  R
 F   0 .2
EC  [0,1]
EC
EC  0.6
Hartree
No interaction
Hartree
Hartree Fock

2C ( ) / e
* energy
scaled
with 

2C ( ) / e2
2
symmetric
L  R
 F  0.5
asymmetric
EC  [0,1]
EC
R / L  9
 F   0 .2
EC  0.6
Hartree
Fock
No interaction
Hartree
Hartree Fock


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Summary
“The noise is the signal” - R. Landauer
The noise spectrum exhibits steps and dips
o The steps
quantum dot resonances
o The dips
energy difference between the resonances
contain information on the coupling asymmetry
 The interaction effect
t
o Shifts levels to preserves neutrality
o Shifts noise steps and dips
 Fock effect
o Small quantitative effect – modifies the width of the levels
Thanks
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