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MIXED, CHARGE AND HEAT NOISES
IN THERMOELECTRIC NANOSYSTEMS
A. Crépieux1 and F. Michelini2
1 – Centre de Physique Théorique, CNRS & Aix-Marseille University, France
2 – IM2NP, CNRS & Aix-Marseille University, France
INTRODUCTION
CURRENT FLUCTUATIONS
“The noise is
the signal”
Rolf LANDAUER
NOISE
  


dt ei t  Iˆ0  Iˆt 
AT EQUILIBRIUM
  0  2kBT0G
 Iˆt   Iˆt   Iˆ
 Gives the linear conductance through the
fluctuation-dissipation theorem
SCHOTTKY RELATION IN THE POISSIONIAN LIMIT (T=0)
  0  e * I
e * / e = Fano factor (1/3, 1, 5/3, 2, ...)
STATISTICS AND DYNAMICS
CROSS-CORRELATOR


4e2
*
*
S23  
Tr s 21s21
s31s31
0
h
S x ,  x  
1  Kc2
eI 0
2
CREPIEUX et al., PRB 67, 205408 (2003)
MARTIN/LANDAUER, PRB 45, 1742 (1992)
 Its sign gives the statistics of the excitations
FINITE FREQUENCY NOISE
G   Re  Y    
Absorption
noise
Emission
noise
CREPIEUX et al., PRB 78, 205422 (2008)
S     S  
2
 The asymmetry of the finite-frequency noise
is related to the ac conductance
HEAT CURRENT NOISE
 JJ   


dt ei  t  Jˆ 0  Jˆ t 
 Jˆ t   Jˆ t  Jˆ

Jˆ t   Iˆ E t  Iˆt 
e
Heat fluctuations give information on higher-order cumulant of charge
KINDERMANN/PILGRAM, Phys. Rev. B 69, 155334 (2004)
counting statistics
Finite-frequency symmetrized heat noise
SERGI, Phys. Rev. B 83, 033401 (2011)
Proposal for the detection of single-electron heat transfer statistics
SANCHEZ/BUTTIKER, Eur. Phys. Lett. 100, 47008 (2012)
Fluctuations of heat current emitted from a single-particle source
BATTISTA et al., Phys. Rev. Lett. 110, 126602 (2013)
Heat fluctuations in driven quantum conductor
MOSKALETS, Phys. Rev. Lett. 112, 206801 (2014)
Energy and power fluctuations in ac-driven coherent conductor
BATTISTA et al., Phys. Rev. B 90, 085418 (2014)
PURPOSE OF THIS WORK
 Find the information contained in the correlator mixing charge and heat currents,
and its link with thermoelectric conversion
CHARGE NOISE
 IpqI 

  Iˆp 0 Iˆq t 
p, q = reservoirs
p=q  auto-correlation
p≠q  cross-correlation
dt
HEAT NOISE
CHARGE CURRENT
Iˆp t   eN p

 Iˆp t   Iˆp t  Iˆp
 JpqJ 
  Jˆ p 0 Jˆq t 
dt
HEAT CURRENT
dQ  dE  dN
MIXED NOISES

  Iˆp 0 Jˆq t 

JI
 pq    Jˆ p 0 Iˆq t 

 IpqJ 
dt
dt
GIAZOTTO et al, Rev. Mod. Phys. 78, 217 (2006)
SANCHEZ et al., New J. Phys. 15, 125001 (2013)
Jˆ p t   IˆpE t  
p ˆ
I p t 
e
 Jˆ p t   Jˆ p t   Jˆ p
RESPONSE
LINEAR LINEAR
RESPONSE
RELATIONS BETWEEN NOISES AND CONDUCTANCES
 IppI  2k BT0G
 JppJ

G = electrical conductance
S = Seebeck coefficient
 = thermal conductance
T0 = average temperature
2k BT02~
 IppJ   JppI  2k BT02 SG
S 
V
T
I 0
  ~  S 2T0G
 Fluctuation-dissipation theorem
applies for each kind of noises
KUBO et al., J. Phys. Soc. Jpn. 12, 1203 (1957)
FIGURE OF MERIT
ZT0 
2
S T0G


 
   
IJ 2
pq
 IpqI
JJ
pq
IJ 2
pq
Independent of p and q
 ZT0 is not upper bounded
 
IJ 2
pq
II JJ
 pq  pq
CREPIEUX / MICHELINI, arXiv:1403:8035 (2014)
LITTMANN/DAVIDSON, J. App. Phys. 32, 217 (1961)
Argument of entropy production
1
Cauchy-Swartz inequality
WHAT ABOUT THE NON-LINEAR REGIME ?
 I   G SG  V 
   
  
~
 J   G    T 
 Optimization of the figure of merit does not guarantee a maximum of
thermoelectric efficiency
max  C
1  ZT0  1
1  ZT0  1
C  Carnot efficiency
 One has rather to consider the ratio between electric and thermal powers
WHITNEY’s talk

th
Pout
Pinel
el
Pout
  th
Pin
Voltage to heat conversion
Heat to voltage conversion
SINGLE LEVEL QUANTUM DOT
L
 L , TL
R
0
 R , TR
L, R   F  eV / 2
TL, R  T0  T / 2
I R  I L
JR  JL
IL
JL
  L  R
LANDAUER-LIKE EXPRESSION

 pq
1

h
    p    q 

n
n
 ,  I, J 
F  d
nI  0
nJ  1
F        f L   1  f L    f R   1  f R        1       f L    f R    2
   
2
   0  2   2

    L, R 

f L, R    1  exp
 k T


 B L, R 
1
 This approach allows to study the noises varying V, T, T0, 0 and 
NOISES
GENERAL RELATION
  IpqI    IpqJ    JpqI  0
p, q
p, q
p,q
 Power conservation
  JpqJ  V 2  ILLI
p,q
JˆL  JˆR  V IˆL  Pth  Pel
CHANGE OF SIGN
Charge
noise
Mixed
noise
 Charge conservation
Heat
noise
kBT / 
region of positive sign
region of negative sign
0 /   2
kBT0 /   1


RL   LR
 The heat cross-correlator
can change its sign contrary to
the charge cross-correlator
eV / 
SCHOTTKY REGIME
WEAK TRANSMISSION     1
Tight energy-charge coupling: J R   0   R  I R
NOISES
ESPOSITO et al., Eur. Phys. Lett. 85, 60010 (2009)
I
 ILR
  e IR
J
 JLR
   0   L  J R
J
 ILR
  e J R    0   R  I R
  coth

 0 R
2 k BTR

 0 L
2 k BTL
  1 when T
L, R
0
 Noises are proportional to currents
EFFICIENCY
J
Pth
  el  R
P
IR V
J
 ILR
JR 
e
I
 ILR
IR 
e
J
J
 ILR
 JLR
eV 

 IL  JR
EQUIVALENTLY
 

   
IJ 2
LR
IJ 2
II
JJ
LR
LR  LR
Independent on 
 The efficiency can be written as a ratio of noises
NUMERICAL VERIFICATION
AUTO-RATIO
CROSS-RATIO
EFFICIENCY
k BT /  0  0
k BT0 /  0  0.001
 The efficiency fits with the cross-ratio !
But it has no relation with the auto-ratio
CONCLUSION
MIXED NOISE IN DISTINCT RESERVOIRS
J
 ILR


  IˆL 0 JˆR t  dt
LINEAR RESPONSE REGIME
ZT0 
= a measure of thermoelectric conversion
SCHOTTKY REGIME
 

   
 
   
IJ 2
LR
2
IJ
II
JJ
LR
LR  LR
IJ 2
pq
 IpqI
JJ
pq
IJ 2
pq
CREPIEUX / MICHELINI, arXiv:1403:8035 (2014)
PERSPECTIVES




Mixed noise in a three terminals systems
More realistic transmission coefficient   ,V 
Effect of coulomb interactions
Mixed noise at finite frequency and/or with ac-driven
Thanks to P. Eyméoud, M. Guigou, and R. Whitney