Ecole Thématique du CNRS en Thermoélectricité

Approche Thermodynamique de la Thermoélectricité

Christophe Goupil Laboratoire Interdisciplinaire des Energies de Demain, LIED

Université Paris Diderot

Part one:Electrons and

thermodynamics

Thermoelectricity.

PELTIER (1834) SEEBECK (1823)

“Coupling Ohm’s law and Fourier’s law”

1794 --- 1795: Letter to professor Antonio Maria Vassalli (accademia delle

scienze di torino ) "... I immersed for a mere 30 seconds the end of

such arc into boiling water, removed it and allowing no time for it to cool

down, resumed the experiment with two glasses of cold water. It was then

that the frog in the water started contracting, and it happened even two,

three, four times on repeating the experiment till one end of the iron

previously immersed into hot water did not cool down".

The Volta Story

Alessandro Volta

18 February 1745 – 5 March 1827)

Thermodynamics

What is a good system?

21 QQWP

)1(1

21

2

2

1

121

T

TQW

T

Q

T

QSS

02121

QQWQQ

Fully reversible Fully irreversible

Perfect system means:

conservative transport of the entropy.

Thermal & Electrical coupling.

ThTc1

2

34

• Reversible adiabatic transport of the carriers. (isentropic)

• The convective part contribute to the entropy transport…

…but not the conductive part .( leak)

• => Reduce the conductive part and increase the convective part .

• The electical conductivity shoud be large.

•=> Yes but by increasing mobility only.

• 1: Adiabatic.

• 2: Isothermal.

• 3: Adiabatic.

• 4: Isothermal.

P

Th

Tc

ap(T

h-T

c)

Simple Thermoelectric generator.

+ + + ++ + +

+ ++

N

Th

Tc

an(T

h-T

c)

- - - -- - -

- --

I

I

I

I

Gibbs free energy and entropy.

v1, N1 v2, N2

p

« Entropy per carrier »

Fluid & « Lattice » .

Purely electronic part: “Gas”

(transported entropy)Lattice contribution.

(“boiling walls” )

(Th , Sh)

(Tc , Sh)

( m, N)

Qh

Qc

WTF

Latt

ice

Electronic gas + Lattice Steam + boiling walls

1

1

12

a

el

latel

TZT

ZT = a2

(e + L)· T

Seebeck coefficientElectrical

conductivity

Thermal conductivity

1017 1018 1019 1020 1021

S

insu

lato

rs

met

als

S2

Carrier Concentration

Tota

l K

ZT

semiconductorsL

e

a

ZTmax

PGEC: « Phonon-Glass Electron-Cristal »

The « Graal » of the best TE material.

mH

mCm

En

erg

y

TCTH

eVS = mH-mC =-ea(TH-TC)

eVS

Thermal biasing

I=0

Th

Tc1

2

34

mH

mCm

En

erg

y

TCTH

Vapp=VS

Thermal biasing

I=0TH

eVS = mH-mC =-ea(TH-TC)

mH

mC

m

Energ

y

TCTH

eVS = mH-mC =-ea(TH-TC)

eVS

Vapp= VS +Vpol

eVpol

eVpol

I=0

Thermal & Electrical biasing

Different operating modes: generator of receptor

TH

Operating modes

Generator

(TEG)

Receptor

(Heater)

Receptor

(Cooler)

IRin

VS Vap

p

Ioffe basic TE model.

aITc

Th

Tc

I

a(T

h-T

c)

aITh

K(T

h-T

c)

1/2RI2

1/2RI2

TEG

aITh

I

aITc

K(T

h-T

c)

Tc

Th

1/2RI2

1/2RI2

TEC

I

aITh

aITc

K(T

h-T

c)

Th

Tc

1/2RI2

1/2RI2

TEH

Efficiency or Power for a TEG

ZT=1.5, 15, 30, 300,

Non endoreversible

endoreversible

Thermodynamicof the fluid.

Elastic coefficients

TC

S

C

C

N

TN

N

m2

1

Thermodynamic of the fluid.

Vining 1997TTE

0

2

0

1

a

TC

S

C

C

N

TN

N

m2

1

TT

T

Q

0

2

1

a

Finite time Thermodynamics:Onsager-Callen

Gibbs relation & state equation.

Gibbs relation.

The Fi are potentials of the system.

S(Xi) contains all the information of the system.

Potentials,

1/T is the conjugate of E.

mi /T is the conjugate of N.

Affinities, Generalized forces

2 subsystems exchanging Xi adn X’ i at constant X0i.

Total S is the sum of the entropy of the 2 subsystems.

Fi =0, S extremal

at equilibrium.

« The forces are given by the gradient of the Potentials ».

Potentials Affinities, forces

Fluxes, entropy and linear response.

Flux of the quantity Xi.

Entropy production.

Linear response.

Matrix of

kinetic coefficients.

Generalized

Force.

Coupled fluxes: the Onsager model.

Generalized

Forces.

• Linear & Stationary transport means dissipation.

Remark: Quasi-static thermodynamics

Time constants.

« Instantanate entropy »

Out of equilibrium description.

Yes, but:

• Quasi-staticStationary cond.

• Minimal entropy production.

• Electronic fluid description.

The thermoelectric cells should be large enough to consider:

• Irreversible processes (avoid microreversibility).

• Slow relaxation time compared to microscopic relaxation time.

Jin

JEin

Jout

JEout

T

QdS

QS

T

JJ

QS

What can we do with that?

Current and heat fluxes.

WQdE NeEQ JJJ

m

!

NJeJ

WQdE NeQE JJJ

m

!

Decoupled processes:

Isothermal case: Ohm’s law

J=0, or Dm=0 : Fourier’s law

)1(2 TZT elecJJTE a

Coupled processes.

J=0 : Seebeck effect Isothermal : Peltier effect

There is no « specific» Seebeck or Peltier, nor Thomson, coefficients.

These are only expression of the same underlying physic

under specific thermodynamic condtions, isothermal, no current…

The entropy per carrier is a measurement

of the quality of the electronic fluid

to carry the entropy (remove or add).

The « Entropy per carrier ».

Kinetic coefficients & transport.

===

Are there any more familiar expressions for this?

The conductivity matrix.

The Seebeck coefficient IS the coupling parameter.

Heat &Entropy fluxes

Heat transformation per unit volume.

Energy conservation:

Carriers conservation:

Heat transformation contributions.

Peltier-Thomson & Co.

Isothermal case: Peltier

Thermal gradient case: Thomson

Kelvin relations:

Entropy production per unit volume.

The Wiedeman-Franz Law.

Metal

Non Degenarate Semi-Conductor

Lorentz number:

22

11

22

0

2211

122

22

11

22

12

1

TeL

L

T

LL

L

TeL

L

T

LT

J

T

J

Lorentz number and ZT

The lattice contribution is pure loss!

Part twoSystem optimization

The CNCA engine

Perfect engine? Thot

Tcold

W

Qin

Qout

hot

cold

in

CT

T

Q

W 1

No power

BUT infinite time do produce W.

Why ? Reversible also means acausal. No defined startup conditions!

Solution ? Modification of the boundary conditions.

• J. Yvon, The saclay Reactor: Two Years of Experience in the Use of a Compresed gas as a Heat Transfer Agent, Proceedings of the International Conference on the Peaceful Uses of Atomic Energy (1955)

• P. Chambadal Les centrales nucléaires. Armand Colin, Paris, France, 4 1-58, (1957)

• I.I. Novikov, Efficiency of an Atomic Power Generation Installation, Atomic Energy 3 (1957)• F.L. Curzon & B. Ahlborn, Efficiency of a Carnot Engine at Maximum Power Output, Am. J. Phys. 43 (1975)

Endoreversible

Thot

Tcold

Work

P

ηηCA ηCarnot

Pmax

Finite Time Thermodynamics

FTT

Thot

Tcold

Power

Endoreversible

hot

cold

in

CT

T

Q

W 1

hot

cold

in

CAT

T

Q

W

1

Generator (TEG)Receptor (Heater) Receptor (Cooler)

I

I(V) response:

0

E

E

=>

ZT

I

II

CC

TEG 1)( 0

ZTE 10

General model: presentation

ZT

I

IKIK

CC

TEG 1)( 0

2

0

1

.. .Q hM cM

I VR R

I T TT TK

R R

a

a a

D

0. .( )Q hM cM

advectionconduction

I T I K T Ta

( )oc hM cMV T Ta

Y. Apertet, H. Ouerdane, O. Glavatskaya, C. Goupil et Ph. Lecoeur, EPL 97 (2012)

Effective thermal conductance !

Force-Flux :

2

0 ( )

adv

Q hM cM

load

K

TI K T T

R R

KTEG

a

General model: Onsager description

Conduction

Convection

Is a function

of Rload!

Thevenin model:

Y. Apertet, et al. EPL 97 (2012)

( )oc hM cMV T Ta

2

0 0

' '

contactoc

contact contact

K TV T I

K K K K

V Roc

aa D

General model: resulting picture

ZT

I

IKIK

CC

TEG 1)( 0

For givenThermal contacts

M. Freunek et al., J. Elec. Mat. 38 (2009)K. Yazawa et A. Shakouri, JAP 111 (2012)

See also:

Electric adaptation Thermal adaptation

0 1

1

contact

load

K K ZT

R R ZT

The thermal adapatation is fundamentalfor correct working conditions!

Power

Y. Apertet, et al. EPL 97 (2012)

Special thanks to

• Henni Ouerdane

• Yann Apertet

• Philippe Lecoeur

• Aurélie Michot

• Olga Glavatskaya

• Eckhard Müller

• Knud Zabrocki

• Wolfgang Seifert

• Jeffrey Snyder

• Cronin Vining

Dilbert 10-10-1993

1. Rowe, D.M., Ed. CRC Handbook of Thermoelectrics: Macro to Nano; RC:

Boca Raton, FL, USA, (2006).

2. Seebeck, T.J. Ueber den Magnetismus der galvanischen Kette. Technical

report for the Royal Prussian Academy of Science: Berlin, Germany, (1821).

3. Peltier, J.C.A. Nouvelles expériences sur la caloricité des courants électrique.

Annales de Chimie et de Physique, 56, 371---386 , (1834)

4. Thomson, W. On the Mechanical Theory of Thermo-electric Currents. Trans.

R. Soc. Edinburgh: Earth Sci. 3, 91---98, (1851)

5. Onsager, L. Reciprocal Relations in Irreversible Processes. I. Phys. Rev. 37,

405---426, (1931). Onsager, L. Reciprocal Relations in Irreversible Processes.

II. Phys. Rev. 38, 2265---2279, (1931).

6. Callen, H.B. The Application of Onsager's Reciprocal Relations to

Thermoelectric, Thermomagnetic, and Galvanomagnetic Effects. Phys. Rev.

1948, 73, 1349--1358. Callen, H.B. On the theory of irreversible processes.

PhD thesis, Massachusetts Institute of Technology - (M.I.T.), Cambridge, MA,

USA, (1947)

7. de Groot, S.R. Thermodynamics of Irreversible Processes; North-Holland

Publishing Company: Amstedam, The Netherlands, 1963.

General Bibliography

1. Müller E. , Zabrocki K. , Goupil C., Snyder G.J., and W. Seifert. Functionally

graded thermoelectric generator and cooler elements. In D.M. Rowe, editor,

CRC Handbook of Thermoelectrics: Thermoelectrics and Its Energy

Harvesting, Vol. 1, Chapter 4. CRC Press, Boca Raton, FL, 2012.

2. Vining, C.B. The thermoelectric process. In Materials Research Society

Symposium Proceedings: Thermoelectric Materials - New Directions and

Approaches; Tritt, T., Kanatzidis, M., Lyon, H.B., Jr., Mahan, G., Eds.;

Materials Research Society: Warrendale, PA, USA; pp. 3---13 (1997)

3. Snyder, G.J.; Ursell, T.S. Thermoelectric Efficiency and Compatibility. Phys.

Rev. Lett. 91, 148301, (2003)

4. Goupil, C. Thermodynamics of the thermoelectric potential. J. Appl. Phys.

106, 104907, (2009)

5. Ioffe, A. Semiconductor Thermoelements and Thermoelectric Cooling;

Infosearch, ltd.: London, UK, (1957)

6. Curzon, F.; Ahlborn, B. Efficiency of a Carnot engine at maximum power

output. Am. J. Phys. , 43, 22---24, (1975)

7. Andresen, et al. Thermodynamics in finite time: Extremals for imperfect heat

engines. J. Chem. Phys. 66, 1571---1577, (1977)

8. Apertet Y. et al. Physical Review E 85, 041144 (2012)

9. Apertet Y. et al. Europhysics Letters 97, 28001 (2012)

Specific Bibliography

Addendum

Mesoscopic version

CouplingY Apertet et al. EPL 97 (2012) N. Nakpathomkun et al. PRB 82

(2010)

Macroscopic Mesocopic

Macrososcopic

• One thermodynamic fluid: => ZT

• One engine

• Two heat exchangers

• Two reservoirs

• Strong coupling, or possible leakage

TcoldThot

Power

K0

Mesoscopic• One thermodynamic fluid: => ZT?

• One engine?

• Two heat exchangers?

• Two reservoirs?

• Strong coupling or possible leakage?

?

mH

mC

m

TCTH

eVpol

I=0

Thermal & Electrical biasing

1 2

Ok if one isolated level

TH

T(E)

Coupling & Broadening

• One thermodynamic fluid: => ZT? NO because ZT=f(E)

• One engine? NO

• Two heat exchangers?

• Two reservoirs?

• Strong, or not, coupling possible?

To be considered together

YES for the lattice and

broadening means dissipation