Tammy HumphreyDepartment of Theoretical Physics, University of Geneva*Email: tammy.humphrey@unsw.edu.au Web: www.humphrey.id.au

Thermoelectrics: Thermoelectrics: Reversibility and Reversibility and

Efficiency at Maximum Efficiency at Maximum powerpower

Outline

• Background

• The Physics of Thermoelectrics

• The Thermodynamics of Thermoelectrics

• Comparison to other solid state energy converters and to Carnot cycle

• Efficiency at Maximum power

Brief history of thermoelectrics

• 1823 – Seebeck and Carnot• 1835 – Peltier effect• 1850’s – Kelvin relations and entropy • 1931 – Onsager relations

• 1950’s – Development of Bi2Te3 thermoelectric coolers

• 1993 – Beginning of investigation of low-dimensional thermoelectrics

• 2001 to Present – Experimental development of nanostructured thermoelectrics with significantly higher efficiency than Bi2Te3

Physics underlying thermoelectricsClosed Circuit

I

V0

Open Circuit

Underlying thermodynamics…

Open circuit

Finite thermal conductivity at open circuit means that heat is consumed but no work is done.

Zero efficiency at the crossover from power generation to refrigeration is the hallmark of an irreversible HE.

What is the limiting ‘electronic’ efficiency of a thermoelectric power generator/refrigerator?

First: Heat transfer in Carnot cycle

Feynman on reversible heat engines…The Feynman lectures on Physics, Chapter 44-3

“We need to find an analog of frictionless motion: heat transfer whose direction we

can reverse with only a tiny change. If the difference in temperature is finite,

that is impossible…”

Heat transfer in thermoelectrics

Carnot Efficiency requires reversible electron transport

Reversible electron transport requires equilibrium

“No thermoelectric device can ever reach Carnot efficiency” H. Littman and B. Davidson, J. Appl. Phys., 32 (2) 217 (1961).

Physics of reversible thermoelectrics

One energy where the effect of the temperature gradient cancels with that of the electrochemical potential gradient

One energy at which current reverses:

(Carnot limit)

Physics of reversible thermoelectrics

Constant occupation of states = Equilibrium

(despite temperature and electrochemical potential gradients)

“Reversible thermoelectric nanomaterials”, T. E. Humphrey and H. Linke, Phys. Rev. Lett. 94, 096601 (2005)

Interestingly,

Other heat engines achieve reversibility in the same way…

Carnot efficiency in thermionic devices

T. E. Humphrey, R. Newbury, R. P. Taylor and H. Linke “A Reversible quantum Brownian heat engine for electrons” Phys. Rev. Lett. 89, 116801 (2002)

(Carnot limit)

Solar Cells and LEDs

Finite thermal conductivity between the sun and the cell

at open circuit means that heat is consumed but no

work is done.

Irreversible heat engine

Loss Mechanisms in solar cells

1) Non-absorption of below band-gap photons

2) Lattice thermalisation losses

3) & 4) Junction and contact resistance losses

5) Recombination losses

Reversible thermophotovoltaics

Reversibility achieved at open circuit voltage eVOC, when a filter is used to limit photons exchanged to those with energy equal to the bandgap, EG. Then:

Efficiency of energy conversion is:

(Carnot limit)

P. T. Landsburg and G. Tongue, J. Appl. Phys., 51, R1 (1980)

Thermally pumped laser

Efficiency at population inversion:

H. E. D. Scovil and E. O. Schulz-DuBois, Phys. Rev. Lett., 2, 262 (1959)

Summary of the talk so far:

• Thermoelectric energy conversion can occur reversibly if particle transport is energy selective (i.e. heat transfer can be non-isothermal but still isentropic)

• This is fundamentally different from the situation in cyclic heat engines such as the Carnot cycle (where isothermal heat transfer is essential for reversibility)

• A range of electronic and photonic heat engines share with thermoelectric devices the same mechanism for reversible operation

What about Maximum power?

Finite time thermodynamicsPoints out that:

1) In practice, it is never desirable to operate a heat engine reversibly, as in this limit power output must be zero

2) The difference between the Carnot limit and the actual efficiency of a practical heat engine does not therefore represent a ‘true’ measure of the efficiency gain which might be achieved with further optimization

FFT asks:

1) What is the efficiency of a heat engine which is optimized to produce maximum power rather than maximum efficiency?

2) How general is this result? Is it applicable to a wide range of different heat engines?

- The Curzon Ahlborn ‘endoreversible’ model

•For Carnot cycle and Fourier law heat transfer [1]:

[1] F.L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975) “Efficiency of a Carnot engine at maximum power output”

Finite power production in the ‘Carnot’ cycle

Maximum power in thermoelectrics

“number of electrons”

EnergyE0

Efficiency at maximum power

0 0.2 0.4 0.6 0.8 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1-

MP

/ C

Ratio of efficiency at MP to Carnot efficiency

1-

‘thermionic’

-function

1D2D

3D

CA effic

AcknowledgementsCollaboration partners:- Ali Shakouri (UCSC)- Mark O’Dwyer (University of Wollongong,

Australia)- Heiner Linke (University of Oregon)

Support has been provided by:- ONR- The Australian research council- A Marie Curie Incoming International Fellowship

from the European Commission