IN A THERMOELECTRIC QUANTUM DOT A. Crpieux F. Michelini Marseille, - PowerPoint PPT Presentation

HEAT AND CHARGE CURRENT FLUCTUATIONS IN A THERMOELECTRIC QUANTUM DOT A. Crépieux F. Michelini Marseille, France

THERMOELECTRICITY Seebeck effect Peltier effect Thomson effect 1821 1834 1851 ELECTRICITY ↔ HEAT

APPLICATION Thermocouple Cool water fountain Thermoelectric generator   6 %

APPLICATION   2 %

APPLICATION

FIGURE OF MERIT HER EREM EMANS et et al al. Nature Nan anotechnologies 8, 471 471 (2013 2013) → New fields of research : Nanothermoelectricity / Quantum Thermoelectricity

LINEAR RESPONSE         Onsager relation V J I G SG V            S       Charge current    ~ T I     T 0 S         J G T I 0 T 0 ~  Heat current    2 S T G 0 Wiedemann-Franz law   1 ZT 1 2   2 2    S GT k 0 ZT   B 0 max C   0  2 1 ZT 1 GT 3 e 0 0 OUTSIDE THE LINEAR RESPONSE        I G SG V              The figure of merit is no longer ~          J G T the adequate quantity to quantify thermoelectricity   1 ZT 1    0 max C   1 ZT 1 0

QUESTION Can noise quantifies the thermoelectric conversion ?

MIXED NOISE      dt  ˆ ˆ    I J S I 0 J t pq p q   CHARGE CURRENT HEAT CURRENT         ˆ ˆ ˆ ˆ ˆ ˆ       I t I t I J t J t J p p p p p p    ˆ          I t e N ˆ ˆ ˆ   p E p p J t I t I t p p p e    dQ dE dN p p p p G R  R , T R G L e 0  L , T L I I R L J J L R

VERY FEW STUDIES ON MIXED NOISE France cesco sco Giaz azot otto, , Tero T. Heikkil ilä, , Arttu Luukanen, Alexan ander r M. Savin, and Jukka Jukka P . Pekola

SYSTEM G R  R , T R G L e 0  L , T L I I R L J J L R          e  e  e   H c c c c d d V c d h . c . k k k k k k 0 k k                    k L k R  p  L , R  k  p       reservoir L reservoir R dot dot reservoir transfer METHOD AND ASSUMPTIONS  Mixed noise expressed in terms of two-particles Keldysh Green’s functions  Non- interacting system → Wick’s theorem  Dyson equation of motion for the dot Green’s function  Fourier transform  Wide-band approximation

RESULTS LANDAUER-LIKE EXPRESSIONS   e              f Fermi Dirac distributi on function e e e  e e T L, R I d f f   L L R   e  h T transmissi on coefficien t              1 e e   e  e e T J d f f BUTCHER, JPCM 2, 2, 48 4869 69 (19 (1990 90) L L L R   h ZERO-FREQUENCY NOISES 2 e       e d e I I S F Charge noise pq   h      e    e   e e I J S F d Mixed noise pq q   h       1     e   e   e e J J S F d Heat noise pq p q   h                               e  e e  e  e  e  e  e e  e 2 T T T F f 1 f f 1 f 1 f f L L R R L R

 Conservation rules  Mixed noise at equilibrium  Mixed noise far from equilibrium

CONSERVATION RULES NUMBER OF ELECTRONS IS CONSERVED ˆ ˆ        N N Cste I I 0 I I I L T L R L R L TOTAL HEAT IS NOT CONSERVED Contact resistance dissipation ˆ ˆ      Q Q Cste J J 0 L R L R   J J J R L L POWER CONSERVATION ˆ ˆ    th  el J J V I P P L R R TOTAL CHARGE AND MIXED NOISES       I I I J J I S S S 0 pq pq pq p , q p , q p , q POWER FLUCTUATIONS CONSERVATION           dt   ˆ ˆ  ˆ ˆ   th th  el el J J 2 I I S S V P t P 0 dt P t P 0 pq LL     p , q

AT EQUILIBRIUM (linear response) RELATIONS BETWEEN NOISES AND CONDUCTANCES G = electrical conductance  I I S 2 k T G pp B 0 S = Seebeck coefficient     = thermal conductance I J J I 2 S S 2 k T SG ~  pp pp B 0    2 S T G T 0 = average temperature ~ 0   J J 2 S 2 k T pp B 0 → Fluctuation -Dissipation Theorem applies for any kind of noises KUBO et al., J. Phys. Soc. Jpn. 12, 1203 (1957) FIGURE OF MERIT   2 I J Independent of p and q 2 S S T G   pq 0 ZT    0 2  I I J J I J S S S CREPIEUX / MICHELINI, JPCM 27, 015302 (2015) pq pq pq

FAR FROM EQUILIBRIUM   e  T SCHOTTKY REGIME 1 NOISES    I I S C e I e   e       LR R C coth 1 when T 0 0 R 0 L     e   I J L , R 2 k T 2 k T S C e J C I B R B L LR R 0 R R    e   J J S C J → Noises are proportional to currents LR 0 L R EFFICIENCY th I J J J P J I I S S S I J S       R  LR LR LR eV I R LR J R el V I e C I C J C P e C R L R EQUIVALENTLY   2 Does not depend on C I J S   LR   2  → Thermoelectric efficiency can be written as a ratio of noises I J I I J J S S S LR LR LR CREPIEUX / MICHELINI, JPCM 27, 015302 (2015)

NUMERICAL TEST AUTO-RATIO CROSS-RATIO EFFICIENCY e  k T / 0 B 0 e  k T / 0 . 001 B 0 0 → The efficiency fits with the cross-ratio ! It has no relation with the auto-ratio

CONCLUSION      dt  ˆ ˆ    I J S MIXED NOISE I 0 J t pq p q   allows to quantify thermoelectric conversion   2 I J S  In the linear response regime pq ZT   0  2 I I J J I J S S S pq pq pq   2 I J S   LR   In the Schottky regime 2  I J I I J J S S S LR LR LR

OPEN PROBLEMS  Coulomb interactions, phonons  Mixed noise for ac-driven  Efficiency fluctuations  Mixed noise in a 3-terminals thermoelectric systems  Measurement of mixed noise WHITNEY, PRB RB 91 91, , 11 1154 5425 25 (20 (2015 15)

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