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 1821 Peltier effect 1834 Thomson effect 1851

ELECTRICITY ↔ HEAT

APPLICATION

% 6  

Thermocouple Cool water fountain Thermoelectric generator

APPLICATION

% 2  

APPLICATION

→ New fields of research : Nanothermoelectricity / Quantum Thermoelectricity

FIGURE OF MERIT

HER EREM EMANS et et al al. Nature Nan anotechnologies 8, 471 471 (2013 2013)



2

GT S ZT 

2 2 2

3e k GT

B

  

Wiedemann-Franz law

LINEAR RESPONSE

1 1 1 1

max

     ZT ZT

C

 

OUTSIDE THE LINEAR RESPONSE

The figure of merit is no longer the adequate quantity to quantify thermoelectricity

                            T V G SG G J I  ~

1 1 1 1

max

     ZT ZT

C

 

                            T V G SG G J I  ~

S T0  

Onsager relation

G T S

2

~   

Charge current Heat current



   

I

T V S

 

 

T

I J

QUESTION Can noise quantifies the thermoelectric conversion ?

MIXED NOISE

    dt

t J I

q p



  

 ˆ ˆ

J I pq

  S

     

p p p p p

N e t I I t I t I      ˆ ˆ ˆ ˆ 

         

p p p p p p E p p p p p

dN dE dQ t I e t I t J J t J t J          ˆ ˆ ˆ ˆ ˆ ˆ

CHARGE CURRENT HEAT CURRENT

e0  R , TR  L , TL GR GL

L

I

L

J

R

J

R

I

VERY FEW STUDIES ON MIXED NOISE

France cesco sco Giaz azot

• tto,

, Tero T. Heikkil ilä, , Arttu Luukanen, Alexan ander r M. Savin, and Jukka Jukka P . Pekola

e0  R , TR  L , TL GR GL

L

I

L

J

R

J

R

I

                       

transfer reservoir dot dot R reservoir L reservoir

. .

,



    

   

       

c h d c V d d c c c c H

R L p p k k k k R k k k k L k k k

e e e

METHOD AND ASSUMPTIONS SYSTEM

• 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

       

e e e e T

R L L

f f d h e I   

  

         

e e e  e e T

R L L L

f f d h J    

  

1

 



  

  e e d F h e

pq 2 I I

S

                             

2

1 1 1 e e e e e e e e e e

R L R R L L

f f f f f f F        T T T

   



  

   e e  e d F h e

q pq J I

S

     



  

    e e  e  e d F h

q p pq

1

J J

S

LANDAUER-LIKE EXPRESSIONS ZERO-FREQUENCY NOISES

BUTCHER, JPCM 2, 2, 48 4869 69 (19 (1990 90)

   

t coefficien

• n

transmissi function

• n

distributi Dirac Fermi f

R L,

   e e T

Charge noise Heat noise Mixed noise

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

CONSERVATION RULES

, I J pq , J I pq , I I pq

  

  

q p q p q p

S S S

        dt

P t P dt P t P V

el el th th

q p

  

     

   ˆ ˆ ˆ ˆ

I I LL 2 , J J pq

S S

NUMBER OF ELECTRONS IS CONSERVED TOTAL CHARGE AND MIXED NOISES

ˆ ˆ     

R L T L

I I Cste N N

el th

P P I V J J

R R L

ˆ ˆ     POWER CONSERVATION POWER FLUCTUATIONS CONSERVATION TOTAL HEAT IS NOT CONSERVED

L R

I I  

ˆ ˆ     

R L R L

J J Cste Q Q

L R

J J  

L

I

L

J

Contact resistance dissipation

AT EQUILIBRIUM (linear response)

CREPIEUX / MICHELINI, JPCM 27, 015302 (2015)

RELATIONS BETWEEN NOISES AND CONDUCTANCES

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

 ~ 2 2 2

2 J J pp 2 I J pp J I pp I I pp

T k SG T k G T k

B B B

     S S S S

   

2 J I pq J J pq I I pq 2 J I pq 2

S S S S     G T S ZT G T S

2

~   

Independent of p and q

FIGURE OF MERIT

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

→ Fluctuation-Dissipation Theorem applies for any kind of noises

FAR FROM EQUILIBRIUM

EFFICIENCY    

R L R R R R

J C I C J e C I e C  e  e      

J J LR J I LR I I LR

S S S

R R

I V J P P

el th

  

SCHOTTKY REGIME → Noises are proportional to currents

 

when 1 coth

, 2 2

   

  R L T k T k

T C

L B L R B R

 e  e

C J C I eV

R L J J LR J I LR

S S  

NOISES → Thermoelectric efficiency can be written as a ratio of noises

C e J R

J I LR

S 

C e I R

I I LR

S 

EQUIVALENTLY

   

J J LR I I LR 2 J I LR 2 J I LR

S S S S   

Does not depend on C

 

1  e T

CREPIEUX / MICHELINI, JPCM 27, 015302 (2015)

NUMERICAL TEST

AUTO-RATIO CROSS-RATIO EFFICIENCY → The efficiency fits with the cross-ratio ! It has no relation with the auto-ratio

001 . / /   e e T k T k

B B

CONCLUSION

MIXED NOISE allows to quantify thermoelectric conversion

   

2 J I pq J J pq I I pq 2 J I pq

S S S S   ZT

   

J J LR I I LR 2 J I LR 2 J I LR

S S S S   

In the linear response regime In the Schottky regime

    dt

t J I

q p



  

 ˆ ˆ

J I pq

  S

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)

Thank you for your attention !