mixed, charge and heat noises in thermoelectric …crepieux/stock/crepieux.pdf · mixed, charge and...
TRANSCRIPT
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,