Thermoelectrics: Reversibility and Efficiency at Maximum power Tammy Humphrey Department of Theoretical Physics, University of Geneva *Email: tammy.humphrey@unsw.edu.au Web: www.humphrey.id.au

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 thermoelectrics Closed Circuit I Open Circuit V0

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

Physics of reversible thermoelectrics One energy at which current reverses: (Carnot limit) 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)

Other heat engines achieve reversibility in the same way… Interestingly, Other heat engines achieve reversibility in the same way…

Carnot efficiency in thermionic devices (Carnot limit) 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)

Solar Cells and LEDs 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 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

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)

What about Maximum power? 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 thermodynamics Points out that: In practice, it is never desirable to operate a heat engine reversibly, as in this limit power output must be zero 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: What is the efficiency of a heat engine which is optimized to produce maximum power rather than maximum efficiency? How general is this result? Is it applicable to a wide range of different heat engines?

Finite power production in the ‘Carnot’ cycle - 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”

Maximum power in thermoelectrics “number of electrons” Energy E0

Efficiency at maximum power 0.2 0.4 0.6 0.8 1 0.3 0.5 0.7 0.9 1- t h MP / C Ratio of efficiency at MP to Carnot efficiency 1- ‘thermionic’ -function 1D 2D 3D CA effic

Acknowledgements Collaboration 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