Laboratoire de Physique et Modélisation des Milieux Condensés Univ. Grenoble & CNRS, Grenoble, France
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1 Laboratoire de Physique et Modélisation des Milieux Condensés Univ. Grenoble & CNRS, Grenoble, France The most efficient quantum thermoelectric at finite power output Robert S. Whitney Phys. Rev. Lett. 112, (2014) & preprint arxiv: (all the details) Marseille November 2014
2 OVERVIEW Question about textbook thermodynamics Some existing thermoelectric machines Origin of thermoelectricity Scattering theory for thermoelectric thermodynamics (1st law, 2nd law, etc) Most efficient thermoelectric at finite power output
3 QUESTION about textbook THERMODYNAMICS Efficiency η = power output heat input Carnot s upper bound: η η Carnot 1 T cold / Thot Carnot efficiency reversibility zero power output science.howstuffworks.com
4 QUESTION about textbook THERMODYNAMICS Efficiency η = power output heat input Carnot s upper bound: η η Carnot 1 T cold / Thot Carnot efficiency reversibility zero power output science.howstuffworks.com What does zero mean? Is there an upper bound more strict than η Carnot at finite power?
5 SOME EXISTING THERMOELECTRIC MACHINES HEAT FLOW CHARGE FLOW POWER GENERATION or REFRIGERATION Purely solid state No MOVING parts
6 SOME EXISTING THERMOELECTRIC MACHINES A.F. Ioffe (1958) Inst. Semicond. Leningrad T T ZnSb thermoelectric 2 Watts (80-90 Volts) HOT = 572 K = 305 K Efficiency η < 2% when Carnot limit would be η Carnot = 47%
7 SOME EXISTING THERMOELECTRIC MACHINES A.F. Ioffe (1958) Inst. Semicond. Leningrad ZnSb thermoelectric 2 Watts (80-90 Volts) T T HOT = 572 K = 305 K Heat = 5kg plutonium α-decay Power output = 120W Efficiency η = 6% Efficiency η < 2% when Carnot limit would be η Carnot = 47%
8 ORIGIN of THERMOELECTRICITY
9 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric
10 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric
11 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric
12 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric
13 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric
14 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric
15 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric
16 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric
17 THERMOCOUPLE CIRCUIT HOT other thermoelectric Load work thermoelectric Ê Î ÊËÁ Ä Carnot efficiency Mahan,Sofo (1996). Humphrey,Linke (2005)... but no power
18 SCATTERING THEORY thermodynamics
19 Landauer-Büttiker scattering theory for charge and heat flow Heat current: J L = de ( ) h (E ev L) T RL (E) f L (E) f R (E) transmission T RL (E) = probability to go L R at energy E [ [ ]] 1 E ev Fermi-function f L (E) = 1+exp L k BT L Treats e-e interactions in mean-field manner (Hartree-like) self-consistently Christen-Büttiker (1996)
20 Scattering theory = Thermodynamics Bruneau Jakšić Pillet, Commun. Math. Phys. 319, 501 (2013) Energy Conservation 1st law thermodynamics HEAT WORK R.W., PRB 87, (2013) Proved scattering theory 2nd law thermodynamics... using Clausius definition of entropy, S = heat T
21 OPTIMIZING EFFICIENCY for ÁÎ Æ POWER OUTPUT
22 OPTIMIZING EFFICIENCY for ÁÎ Æ POWER OUTPUT
23 OPTIMIZING EFFICIENCY for ÁÎ Æ POWER OUTPUT
24 OPTIMIZING EFFICIENCY for ÁÎ Æ POWER OUTPUT Variables: height of each slice & bias, V Constraint : power= P
25 OPTIMIZING EFFICIENCY for ÁÎ Æ POWER OUTPUT Variables: height of each slice & bias, V Constraint : power= P
26 OPTIMAL TOP-HAT WIDTH Transcendental equation for top-hat position and width zero power output increasing power output max. power output transmission transmission transmission energy energy energy
27 POTENTIAL REALIZATION Make chain of sites (tight-binding model) = states form a band
28 Max. EFFICIENCY for GIVEN POWER Carnot : η Carnot = 1 T Hot /T Cold Efficiency 0 Power output, P
29 Max. EFFICIENCY for GIVEN POWER Carnot : η Carnot = 1 T Hot /T Cold Efficiency 0 Power output, P Quantum bound (qb) on power P qb A0π2 6h N ( ) 2 k B T Hot k B T Cold with A / quantum since N cross-section Fermi wavelength
30 Max. EFFICIENCY for GIVEN POWER Carnot : η Carnot = 1 T Hot /T Cold Efficiency ( ) η(p) = η Carnot P 1 α 1 + P qb 0 Power output, P Quantum bound (qb) on power P qb A0π2 6h N ( ) 2 k B T Hot k B T Cold with A / quantum since N cross-section Fermi wavelength intimately related to Pendry (1983) bound on heat/entropy flow
31 Max. EFFICIENCY for GIVEN POWER Is quantum bound on power P qb relevant for applications? For 100W of power from T Hot T Cold 300K with wavelength λ F 10 8 m Requires thermoelectric cross-section > 4mm 2 For 90% of Carnot requires cross-section > 0.4cm 2
32 CONCLUSIONS Scattering theory thermodynamics PRB 87, (2013) Max. efficiency at zero-power (Carnot) is Ð Ð... but max. efficiency at finite-power is ÕÙ ÒØÙÑ given by top-hat transmission PRL 112, (2014) transmission energy Machines capable of Carnot are NOT the best at finite power Other stuff : phonons, relaxation, etc arxiv:
33 === EXTRAS ===
34 PROOF increasingγth slice height,τ γ, will decreasej (increases efficiency) only if 0 > J ( ) ǫγ τ γ = P ev J P P τ γ V primed =d/dv ǫ 1 ev J P = 0 ǫ 0 ev 1 T R /T L = 0 Energy-integrals in J and P are Fermi-functions top-hat transmission ǫ 0 ǫ 1 energy P & J are sums of logs and dilog.-functions ln [ 1+e (ǫ evj)/kbtj ] & Li2 [ e (ǫ ev j)/k BT j ] Getǫ 1 from transcendental eq. for givent L (hot),t R (cold) & power,p
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