A. Crépieux1 and F. Michelini2

1 – Centre de Physique Théorique, CNRS & Aix-Marseille University, France 2 – IM2NP, CNRS & Aix-Marseille University, France

MIXED, CHARGE AND HEAT NOISES

IN THERMOELECTRIC NANOSYSTEMS

ItItI ˆˆˆ

CURRENT FLUCTUATIONS

“The noise is the signal”

Rolf LANDAUER

tIIedt ti ˆ0ˆ

NOISE

AT EQUILIBRIUM Gives the linear conductance through the fluctuation-dissipation theorem GTkB 020

INTRODUCTION

SCHOTTKY RELATION IN THE POISSIONIAN LIMIT (T=0)

Ie*0 = Fano factor (1/3, 1, 5/3, 2, ...) ee /*

CROSS-CORRELATOR

MARTIN/LANDAUER, PRB 45, 1742 (1992)

02

1,

2

IeK

xxS c

CREPIEUX et al., PRB 67, 205408 (2003)

STATISTICS AND DYNAMICS

FINITE FREQUENCY NOISE

Its sign gives the statistics of the excitations

2Re

SSYG

Emission noise

Absorption noise

CREPIEUX et al., PRB 78, 205422 (2008)

The asymmetry of the finite-frequency noise is related to the ac conductance

0Tr4 *

3131*2121

2

23 ssssh

eS

Heat fluctuations give information on higher-order cumulant of charge counting statistics

HEAT CURRENT NOISE

SERGI, Phys. Rev. B 83, 033401 (2011)

BATTISTA et al., Phys. Rev. Lett. 110, 126602 (2013)

KINDERMANN/PILGRAM, Phys. Rev. B 69, 155334 (2004)

tJJedt tiJJ ˆ0ˆ

JtJtJ ˆˆˆ

Finite-frequency symmetrized heat noise

Fluctuations of heat current emitted from a single-particle source

BATTISTA et al., Phys. Rev. B 90, 085418 (2014)

MOSKALETS, Phys. Rev. Lett. 112, 206801 (2014)

SANCHEZ/BUTTIKER, Eur. Phys. Lett. 100, 47008 (2012)

Proposal for the detection of single-electron heat transfer statistics

Heat fluctuations in driven quantum conductor

Energy and power fluctuations in ac-driven coherent conductor

tIe

tItJ E ˆˆˆ

PURPOSE OF THIS WORK

Find the information contained in the correlator mixing charge and heat currents, and its link with thermoelectric conversion

CHARGE NOISE

MIXED NOISES

HEAT NOISE

dttII qp

ˆ0ˆII

pq

dttJJ qp

ˆ0ˆJJ

pq

dttIJ

dttJI

qp

qp

ˆ0ˆ

ˆ0ˆ

IJpq

JIpq

GIAZOTTO et al, Rev. Mod. Phys. 78, 217 (2006) SANCHEZ et al., New J. Phys. 15, 125001 (2013)

ppp

pp

ItItI

NetI

ˆˆˆ

ˆ

ppp

ppE

pp

JtJtJ

tIe

tItJ

dNdEdQ

ˆˆˆ

ˆˆˆ

p, q = reservoirs p=q auto-correlation p≠q cross-correlation

CHARGE CURRENT

HEAT CURRENT

LINEAR RESPONSE

CREPIEUX / MICHELINI, arXiv:1403:8035 (2014)

RELATIONS BETWEEN NOISES AND CONDUCTANCES

KUBO et al., J. Phys. Soc. Jpn. 12, 1203 (1957)

SGTk

Tk

GTk

B

B

B

20

IJpp

JIpp

20

JJpp

0II

pp

2

~2

2

2JI

pqJJ

pqII

pq

2JIpq0

2

0

GTSZT

GTS 02~

1

JJpq

IIpq

2JIpq

Independent of p and q

LINEAR RESPONSE

Fluctuation-dissipation theorem applies for each kind of noises

ZT0 is not upper bounded LITTMANN/DAVIDSON, J. App. Phys. 32, 217 (1961)

Cauchy-Swartz inequality

Argument of entropy production

FIGURE OF MERIT

G = electrical conductance S = Seebeck coefficient = thermal conductance T0 = average temperature

0

IT

VS

WHAT ABOUT THE NON-LINEAR REGIME ?

T

V

G

SGG

J

I

~

11

11

0

0max

ZT

ZTC

Optimization of the figure of merit does not guarantee a maximum of thermoelectric efficiency

One has rather to consider the ratio between electric and thermal powers

WHITNEY’s talk

elin

thout

P

P Voltage to heat conversion

thin

elout

P

P

efficiencyCarnotC

Heat to voltage conversion

LANDAUER-LIKE EXPRESSION

2/, eVFRL

2/0, TTT RL

RL

22

0

2

0 R , TR

L , TL

R L

LI

LJ LR JJ

LR II

SINGLE LEVEL QUANTUM DOT

dF

h

nq

np

1pq

2111 RLRRLL ffffffF

1

,

,, exp1

RLB

RLRL

Tkf

JI ,,

This approach allows to study the noises varying V, T, T0, 0 and

1

0

J

I

n

n

NOISES

The heat cross-correlator can change its sign contrary to the charge cross-correlator

region of positive sign

region of negative sign

2/0

1/0 TkB

/eV

/TkB

LRRL

Mixed noise

Heat noise

Charge noise

0,

IJpq

,

J Ipq

,

I Ipq

qpqpqp

I ILL

2

,

JJpq V

qp

Power conservation

GENERAL RELATION

elthLRL PPIVJJ ˆˆˆ

Charge conservation

CHANGE OF SIGN

EFFICIENCY

RRRLR

RLLR

RLR

IJe

J

Ie

0JI

0JJ

II

VI

J

P

P

R

Rel

th

SCHOTTKY REGIME

WEAK TRANSMISSION 1

Noises are proportional to currents

0when1coth ,2200

RLTkTk

TLB

L

RB

R

R

LR

L

LR

JIeV

JJJI

NOISES

The efficiency can be written as a ratio of noises

eJ LR

R J I

e

I LRR

I I

EQUIVALENTLY

JJ

LRII

LR

2JILR

2JILR

Independent on

Tight energy-charge coupling: RRR IJ 0

ESPOSITO et al., Eur. Phys. Lett. 85, 60010 (2009)

AUTO-RATIO CROSS-RATIO

EFFICIENCY

NUMERICAL VERIFICATION

The efficiency fits with the cross-ratio ! But it has no relation with the auto-ratio

001.0/

0/

00

0

Tk

Tk

B

B

Thanks to P. Eyméoud, M. Guigou, and R. Whitney

MIXED NOISE IN DISTINCT RESERVOIRS LINEAR RESPONSE REGIME SCHOTTKY REGIME

CONCLUSION

2JI

pqJJ

pqII

pq

2JIpq

0

ZT

dttJI RL

ˆ0ˆJI

LR = a measure of thermoelectric conversion

PERSPECTIVES

CREPIEUX / MICHELINI, arXiv:1403:8035 (2014)

Mixed noise in a three terminals systems More realistic transmission coefficient Effect of coulomb interactions Mixed noise at finite frequency and/or with ac-driven

JJ

LRII

LR

2JILR

2JILR

V,